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Efimov Effect
in Pionless Effective Field Theory
and its Application to
Hadronic Molecules
Dissertation
zur Erlangung des
Doktorgrades (Dr. rer. nat.)
der Mathematisch-Naturwissenschaftlichen Fakultat
der Rheinischen Friedrich-Wilhelms-Universitat Bonn
vorgelegt von
Erik Wilbring
aus Troisdorf
Bonn Dezember 2015
II
Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakultatder Rheinischen Friedrich-Wilhelms-Universitat Bonn
1. Gutachter: Professor Dr. Hans-Werner Hammer
2. Gutachter: Professor Dr. Ulf-G. Meißner
Tag der Promotion: 21.1.2016
Erscheinungsjahr: 2016
III
IV
Contents
Preamble 1
1 Introduction and theoretical background 21.1 Quantum chromodynamics and conventional hadrons . . . . . . . . . . . . . . . . 21.2 Exotic hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Effective field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.1 EFT’s for QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Basic scattering theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.1 Some remarks on resonances, bound and virtual states . . . . . . . . . . . 121.4.2 Pole position and effective range expansion . . . . . . . . . . . . . . . . . . 13
2 Universality and the Efimov effect 152.1 Efimov effect for identical bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.1 Three-body scattering amplitude . . . . . . . . . . . . . . . . . . . . . . . 182.2 Efimov physics in cold atoms and halo nuclei . . . . . . . . . . . . . . . . . . . . . 22
3 Efimov effect in a general three particle system 243.1 General dimer states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.1 Dimer flavor wave function . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.2 Spin and isospin part of the dimer wave function . . . . . . . . . . . . . . 273.1.3 Spatial part of the dimer wave function . . . . . . . . . . . . . . . . . . . . 28
3.2 Lagrangian density, vertices and propagators . . . . . . . . . . . . . . . . . . . . . 293.2.1 Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.2 Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2.3 Wave function renormalization constant . . . . . . . . . . . . . . . . . . . 40
3.3 General three-body scattering amplitude . . . . . . . . . . . . . . . . . . . . . . . 413.3.1 Wave function renormalization . . . . . . . . . . . . . . . . . . . . . . . . . 443.3.2 Projection on partial wave amplitude . . . . . . . . . . . . . . . . . . . . . 443.3.3 Spin and isospin projection onto specific scattering channel . . . . . . . . . 45
3.4 Asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4.1 Scale and inversion invariance and decoupling of amplitudes . . . . . . . . 493.4.2 Type 1 systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.4.3 Type 2 systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
V
3.4.4 Type 3 systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4.5 Implementation with Mathematica . . . . . . . . . . . . . . . . . . . . . . 71
4 Efimov effect in hadronic molecules 72
4.1 Established systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.1.1 Three nucleon system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.1.2 KKK system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.1.3 Summary of other established systems . . . . . . . . . . . . . . . . . . . . 77
4.2 Hypothetical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2.1 Hypothetical charm and bottom meson systems . . . . . . . . . . . . . . . 80
4.2.2 More dimers in existing charm and bottom meson systems . . . . . . . . . 84
4.2.3 Hypothetical dibaryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5 Elastic molecule–particle scattering observables 90
5.1 Elastic S-wave Z(′)b –B(∗) scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1.1 Zb–B scattering amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.1.2 Zb–B∗ scattering amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.1.3 Z ′b–B scattering amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1.4 Z ′b–B∗ scattering amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2 Numerical determination of scattering length and phase shift . . . . . . . . . . . . 99
5.2.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6 Conclusion and outlook 105
A Spin and isospin projection operators 107
A.1 Combined spin and isospin projection operators . . . . . . . . . . . . . . . . . . . 107
A.2 Projection onto scattering channel . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
A.3 x, y, z parameters and the 6-J-symbol . . . . . . . . . . . . . . . . . . . . . . . . . 118
A.4 Projection operator summary tables . . . . . . . . . . . . . . . . . . . . . . . . . . 129
B Symmetry factors 136
B.1 Symmetry factor Sij of self-energy diagrams . . . . . . . . . . . . . . . . . . . . . 137
B.2 Symmetry factor Sel of two-body elastic scattering diagrams . . . . . . . . . . . . 139
B.3 Symmetry factor Sijk of dimer–particle scattering diagrams . . . . . . . . . . . . . 141
C Feynman diagrams contributing to d12–P3 dimer–particle scattering 152
D Projection on L-th partial wave 162
E Mellin transform of Legendre functions of the second kind 169
F Three-body scattering amplitude equations 174
VI
G Numerical implementation of scattering amplitudes 205
Bibliography 209
Acknowledgments 218
VII
VIII
Preamble
In the past years various new hadronic states were discovered in the charm and bottom mesonsector by experiments like Belle, BES and Babar. Many of these states fit into the conventional ccor bbmeson spectrum, but a growing number of them does not. Especially, the charged ones whichcan definitely be no conventional hadrons motivate the term ”exotic“ hadrons to classify particleswhich are neither conventional baryons nor conventional mesons. Their substructure is not yetclear due to the often poor experimental knowledge and there are many different interpretationsfor these exotic particles. One very popular idea is the hadronic molecule interpretation whichstates that two ordinary hadrons can form a loosely bound pair similar to the deuteron made of aproton and a neutron. However, in the latter system it is known that after adding a third nucleonthere appears a three particle bound state, the triton. The emergence of such a three-body statecan be explained by the Efimov effect as a low-energy universality phenomenon. The questionwhich will be answered in this work is if there exists also an Efimov effect in systems of hadronicmolecules which scatter off a third meson or baryon. Since the number of candidate systems islarge (and still growing due to ongoing experiments) it is useful to find a general expression thattells us if the Efimov effect is present. Moreover, this expression should only depend on the basicparticle properties like mass, spin and isospin. Such a relation will be derived in this work for ageneric three particle system that can be described in a pionless effective field theory. It will thenbe applied to a set of possible hadronic molecule systems in order to check if the Efimov effectoccurs. The thesis is organized as follows: after an introduction of quantum chromodynamicsand a review of the classification of hadrons we continue with a summary, firstly, of experimentalobservations concerning exotic hadrons, secondly, of the idea of effective field theories and thirdly,of basic scattering theory. In chapter 2 the three identical boson system will be discussed in orderto explain the Efimov effect more quantitatively. With this knowledge it is possible to derivethe mentioned general expression regarding the existence of Efimov physics in a generic threeparticle system. After the detailed derivation in chapter 3 (using the information given in theappendices) the results will be applied to scattering processes where known hadronic moleculesas well as hypothetical ones are involved. This is done in order to either search for the Efimoveffect or – for the hypothetical molecular states – to predict under which conditions an Efimovtrimer is expected. For system without Efimov effect we will calculate in chapter 5 for a selectedexample S-wave observables of the in this case elastic scattering process. The thesis is completedby a summary of the results and an outlook on further work that can be done based on this work.
1
Chapter 1
Introduction and theoreticalbackground
1.1 Quantum chromodynamics and conventional hadrons
Nature is described by four main forces: gravity, electromagnetic, weak and strong force. Whilegravity is up to now not yet – in a well established way – quantized, the latter three are com-bined within the standard model of particle physics. Namely, one uses the concept of quantumfield theories with local gauge invariance (gauge theories) to describe electromagnetic, weak andstrong interactions. On the one hand quantum electrodynamics (QED), i.e. the theory of theelectromagnetic and – with some extensions – also of the weak force (electroweak unification),is perturbative. This is the case because the corresponding running coupling constant is muchsmaller than 1 for a wide energy range. On the other hand the strong force described by quan-tum chromodynamics (QCD) has a coupling constant whose running is much faster. Hence, fora wide energy range, especially in the region of hadron formation, it is of the order of 1 andthus, one cannot use perturbation theory. Apart from this subtlety (which is further discussedin section 1.3) the standard model of particle physics contains twelve fundamental fermions (aswell as twelve anti-fermions) and twelve gauge bosons (the photon γ, W±- and Z0-boson as wellas eight gluons) which are the carriers of the electroweak and strong force. One half of the funda-mental fermions are leptons (electron (e−), muon (µ−), tauon (τ−) and the three correspondingneutrinos νe, νµ, ντ ) and the other half are quarks. As the latter are the constituents of hadronswe focus on them in more detail. The quarks are grouped into three (weak isospin) doublets
(ud
),
(cs
),
(tb
), (1.1)
and named up (u), down (d), strange (s), charm (c), bottom (b) and top (t), also known asquark flavors. Besides the properties summarized in Tab. 1.1 quarks also carry a color chargewith three degrees of freedom (sometimes referred to as ”red“, ”blue“ and ”green“). The colorinteractions are described in terms of quantum chromodynamics whose underlying group is theSU(3)c color gauge group. There are eight gauge bosons in the theory called gluons (g) whichthemselves carry color and anti-color. As already mentioned a bound state made of quarks is
2
d u s c b t
Q – electric charge −13
+23−1
3+2
3−1
3+2
3
I – isospin 12
12
0 0 0 0
I3 – isospin z-component −12
+12
0 0 0 0
S – strangeness 0 0 −1 0 0 0
C – charm 0 0 0 1 0 0
B – bottomness 0 0 0 0 −1 0
T – topness 0 0 0 0 0 1
Table 1.1: Additive quantum numbers of the quarks, from Ref. [2].
called a hadron. Furthermore, we know that due to confinement (i.e. due to the fact that thereis no experimental evidence for free colored particles (e.g. free quarks), see e.g. Ref. [2]) sucha bound state must be color-neutral, i.e. it must be a color singlet which is invariant underSU(3)c transformations. Following Ref. [1] a quark qi with color index i transforms under thefundamental representation 3c of SU(3)c and an anti-quark qi under the 3c. Therefore one canform a SU(3)c invariant object in two ways: firstly, one can combine a quark and an anti-quarkvia qiδij qj so that it is in the 1c (the singlet) of 3c ⊗ 3c = 1c ⊕ 8c. These qiqi objects are calledmesons. Secondly, one can use the total antisymmetric tensor εijk to form a three quark color-neutral object εijk qiqjqk known as baryon or a color singlet three anti-quark state εijk qiqj qk,the anti-baryon. At first glance, it seems that conventional mesons, baryons and anti-baryonsare the only possible hadrons in the standard model. However, one can combine these alreadycolor-neutral objects to color-singlets with more than three quarks. These states are referred toas exotic hadrons and will be discussed in more detail in section 1.2.
It is known that the masses of the three lightest quarks u, d and s (obtained in the MS schemeat a scale of 2 GeV) are [2]
mu = 2.3+0.7−0.5 MeV , md = 4.8+0.5
−0.3 MeV , ms = 95± 5 MeV , (1.2)
and thus relatively close to each other especially compared to the much heavier charm (mc ∼1.3 GeV), bottom (mb ∼ 4.2 GeV) and top (mt ∼ 173 GeV) quarks (for a detailed discussion ofthe methods used to deduce the quark masses see Ref. [2]). Therefore one assumes an approxi-mate SU(3)f flavor symmetry for the three lightest quark flavors. Hence, quantum mechanicsinduce that the bound states of the three lightest quarks will mix. For mesons this motivatestheir representation in a multiplet with 3f⊗ 3f = 1f⊕8f ground states with angular momentumL = 0 classified by charge Q, isospin I and strangeness S. Since the quarks a fermions with spin1/2 the total spin of the corresponding mesons can be either 0 or 1. Thus, there are two nonets(i.e. a singlet plus an octet), one for pseudoscalar (JP = 0−) mesons shown in Fig. 1.1(a) andone for vector mesons (JP = 1−), see Fig. 1.1(b). The former pseudoscalar multiplet contains anisospin doublet with strangeness S = +1 identified with the two pseudoscalar kaons, an isospinanti-doublet with strangeness S = −1 (the pseudoscalar anti-kaons), an isospin triplet of π+, π0,
3
π− without strangeness and in addition the two particles η and η′ with Q = I = S = 0 which aresuperpositions of the SU(3)f flavor octet and singlet states η8 and η1 (see for example Ref. [2,3]).The same pattern is also found in the vector meson nonet where the pseudoscalar kaons arereplaced by their vector counterparts (indicated by a ∗ superscript). The vector iso-triplet getsthe symbol ρ and the two SU(3)f octet / singlet superpositional states with Q = I = S = 0 arenamed ω and ϕ (note, that their mixing is different from that of η and η′ [2, 3]).For the baryons one has to keep in mind that they are fermions, consequently, their wave function
|baryon〉 = |color〉 × |flavor〉 × |spin〉 × |space〉 , (1.3)
must be antisymmetric. Following Ref. [2] it is known that the color wave function is antisym-metric and the spatial part is symmetric for angular momentum L = 0 (ground state). Thus,|flavor, spin〉 must be symmetric, too. Combining both the SU(3)f flavor group of the lightquarks and the SU(2) spin group, one can treat a quark q as an element of a SU(6). Grouptheory states (see e.g. Ref. [2]) that the direct product of three 6’s can be decomposed into fourmultiplets which are either symmetric, antisymmetric or of mixed symmetry:
6⊗ 6⊗ 6 = 56sym ⊕ 70mixed ⊕ 70mixed ⊕ 20antisym . (1.4)
As mentioned above we need a symmetric representation for flavor and spin. Hence, the 56sym isthe right choice. Since it is more common to treat quark flavor and spin separately we decomposethe symmetric 56-plet again into two ground state SU(3)f flavor multiplets where each belongsto a fixed spin quantum number [2]:
56sym = 410f ⊕ 28f , (1.5)
with a superscript (2J + 1) which defines the total spin J of the three quark state. Namely,one finds a spin 3/2 flavor decuplet and a spin 1/2 flavor octet, where the elements in bothare classified by their electric charge Q, strangeness S and isospin I. In the octet (Fig. 1.1(c))one has the nucleon isospin doublet of the proton and neutron as well as a strangeness S = −1iso-triplet (Σ+, Σ0, Σ−) and iso-singlet (Λ) and finally, a double-strange iso-doublet containingthe two Ξ baryons with zero and negative charge. In the decuplet (Fig. 1.1(d)) one finds the ∆resonances and the spin 3/2 states of Σ and Ξ and furthermore the Ω− with strangeness S = −3.
Up to know we have only considered the three lightest quarks, but of course there are also hadronsmade of charm and bottom quarks (note, that due to its very short life time the top quark is notrelevant in hadron formation [2]). In order to take such states into account one can extend theSU(3)f flavor group to a SU(4)f or even a SU(5)f flavor symmetry by adding c and b quarks.However, since charm and bottom quarks are much heavier than u, d and s these symmetriesare badly broken. Hence, the quantum mechanical mixture between a state with e.g. an u quarkreplaced by a c is negligible small. The three dimensional multiplets for the SU(4)f containingthe charm quark can be found for example in Ref. [2].
4
K0 K+
π− π+
K0K−
η
π0
η′
Q
S
I3−1
2+1
2
+1
−1
+1
−1
(a) pseudoscalar mesons
K∗0 K∗+
ρ− ρ+
K∗0K∗−
ω
Q
S
I3−1
2+1
2
+1
−1
+1
−1
φ
ρ0
(b) vector mesons
n p
Σ− Σ+
Ξ0Ξ−
Q
S
I3−1
2+1
2
0
−2
+1
−1
Σ0 Λ
(c) spin 1/2 baryons
Ω−
Ξ∗0Ξ∗−
Σ∗− Σ∗0 Σ∗+
∆− ∆0 ∆+ ∆++
S
Q
I3
0
−2
+12
−12
+1
−1
(d) spin 3/2 baryons
Figure 1.1: SU(3)f pseudoscalar (a) and vector (b) meson nonets as well as SU(3)f spin 1/2baryon octet (c) and spin 3/2 baryon decuplet (d). The particles contain u, d and s quarksand the quantum numbers vary from negative to positive values as indicated by the axes: electriccharge Q, third component of the isospin I3 from left to right and strangeness S from bottom totop.
5
1.2 Exotic hadrons
In the previous section we have argued that mesons, baryons and anti-baryons are the conven-tional color-singlets of the standard model (see Fig. 1.2). However, there are in principle moreallowed states: combinations like qq are indeed forbidden as they are not color-neutral, but thepresent knowledge of QCD does not forbid states made of more than three quarks. In few words:the combination of two SU(3)c invariants is still invariant. Thus, one could construct so-calledexotic hadrons via the combinations qqqq, qqqqq, qqqqq, qqqqqq, qqqqqq and qqqqqq or – in prin-ciple – via combinations with even more quarks, grouped in the same manner. Additionally, alsoqqg quark–gluon combinations or even pure gluonic bound states like gg are not forbidden sincethey are color-neutral. Although all these states are referred to as exotic hadrons there is a moreprecise classification which takes into account their substructure: starting from behind there areon the one hand glueballs (gg) [4–6] and hybrids (qqg) [4, 7] where gluons are explicit degrees offreedom and on the other hand pure quark states with various numbers of quarks and anti-quarkswhich are called tetra-, penta- or hexaquarks in case of four, five or six constituents, respectively.These multi-quark states are compact objects which are tightly bound by the strong force, i.e.by quark–gluon interactions.A different type of exotic hadrons are hadronic molecules whose concept was introduced inRefs. [12–14,56,57]. They are defined as multi-quark states with at least four constituents (qqqq)where the quarks cluster into conventional mesons, baryons or anti-baryons which are clearly se-parated and only bound by the nuclear force, i.e. light-meson (dominantly pion) exchange. Thus,hadronic molecules are extended objects in contrast to the compact tetraquarks, pentaquarks,etc. A sketch of the substructure of exotic hadrons is shown in Fig. 1.3.Obviously, a particle which contains for example four quarks could be either a tetraquark or ahadronic molecule. Moreover, even the decay products of such a particle do – in general – notexclude one or the other substructure. A fully solved QCD would answer the question whichexplanation for a given particle is the right one. However, we are far from understanding thistheory completely. Hence, there is still an intensive discussion about the exact substructure ofthe up to now discovered exotic hadrons.
Meson
q qq
Baryon
q qq
Anti-baryon
Figure 1.2: Conventional hadrons.
1.2.1 Experimental results
Considering the results of a large number of finished or still running experiments like BaBar, Belle,BES or LHCb, it is clear that the mentioned discussion about the substructure of exotic hadronsis not an academic one. The former three experiments are located at so-called B-factories,i.e. e+e− colliders originally built for charmonium (cc) and bottomonium (bb) spectroscopy
6
bound
objectsextended
shallowly
Glueball
gg gg
Hybrid
gg
Hadronic molecules
qq q qq q
qq q
qq q
qq q
qq q
qq q
qq q
Tetraquark
qqqq
qq qq q
Pentaquarks
qq qq q
qq qqqq
qqqq
Hexaquarks
qq qqqq
objects
compactbound
tightly
Figure 1.3: Exotic hadrons.
[15–17]. The latter analyzes the collision of protons or lead nuclei at extremely high center-of-mass energies of up to 8 TeV [18]. Contrary to the expectations the B-factories discovered alarge number of states which do not fit into the scheme of conventional hadrons. Furthermore,some of these states are even charged so that it is clear (because of their definitely present cccontent) that they must contain more than two quarks and thus must be exotic. The experimentsobserved among other states the electrically neutral X(3872) [20] and Y (4260) [21] as well asthe charged Zc(3900) [22–24], Zc(4020) [25–28], Zc(4430) [29–31] and the somewhat controversialstates Z1(4051), Z2(4250) [32] in the charmonium sector. Also in the bottomonium sector twostates were found, the Zb(10610) and the Z ′b(10650) [33]. An overview of the experimentallyobserved XYZ states and their possible substructure in the charmonium and bottomonium sectorcan be found in [34, 35] and most recent in Ref. [19] where Cleven et al. reviewed the particlesabove in full detail.Besides these relatively heavy states without any open strangeness, charm or bottomness thereadditionally are at least two particles, D∗s0(2317) and Ds1(2460), whose masses are only half aslarge and which carry open charm and strangeness. They were seen by BaBar [36] and by CLEO-c [37]. Soon after their discovery it was proposed that they could be D(∗)K molecules [43, 44]
7
rather than conventional mesons with cs quark content (later the molecule interpretation wasalso discussed in Refs. [45–48]). A possible third candidate of this type is the D∗s1(2700) seen byBelle [59] which could be a D1K molecule, however, due to its decay into D0K+ it is more likelya radial excited D∗s0 as already indicated by its name.As last candidates for exotic hadrons we mention the long known resonances a0(980) and f0(980)whose exact substructure is a widely discussed subject. A theoretical discussion of the possiblesubstructure of all these states can be found in Refs. [60–62] and from an experimental view inRef. [63].
1.3 Effective field theory
In section 1.1 we have already noted that QCD is not perturbative in the energy region of hadronformation. The self-energy corrections to the gluons which are present due to the fact that gluonsthemselves carry color, yield a strong coupling constant αs which becomes weaker for higherenergies (asymptotic freedom). In fact, for energies around 1 GeV, αs is of order 1 and henceperturbation theory breaks down (for a review of experimental results concerning the running ofαs see Ref. [2]). It is thus necessary to find a different approach to describe QCD at low energies.Indeed, effective field theories (EFT’s) are a perfectly suitable choice since they are designed toperform calculations in the low-energy regime of quantum field theory (QFT). The basic idea isthat a physical phenomenon at a given energy scale E0 is not affected by the details of high-energyeffects at E E0. A semi-classical example would be the scattering of a low-energy photon withwavelength 500nm off a crystal. Since the wavelength is too large to resolve the lattice structureof the crystal, Bragg-scattering as a high-energy phenomenon is not relevant. Following the ideasof Weinberg [64] one needs to identify two scales in a system to construct an EFT: one low-energyscale m and one high-energy scale Λ which define an expansion parameter m/Λ. It allows toarrange all – in general infinitely many – terms in the effective Lagrangian density which areallowed by the symmetries of the original full theory in powers of this expansion parameter. Aslong as m/Λ < 1 one knows that a term proportional to (m/Λ)n+1 is less important than aterm proportional to (m/Λ)n. In this way one obtains a power-counting scheme which allowsto calculate observables up to a certain order, knowing that the contribution of all higher orderterms is small and thus can be neglected. Moreover, it is possible to use experimental data to fitunknown constants (so-called low-energy constants, LEC’s). These constants are prefactors ofthe terms in the effective Lagrangian, so it is in principle possible to calculate observables to anarbitrary high order as long as enough experimental input is available. However, one subtlety ofEFT’s is that they are not renormalizable because new parameters (the LEC’S) appear in themif one goes to higher orders. In fact, this distinguishes an EFT from conventional perturbationtheory in which independently of the considered order, the number of parameters is fixed in away that the theory can be renormalized. Furthermore, the expansion parameter in an EFT isnot assumed to be fundamental in physics. Consequently, the effective scales can change in away that the expansion parameter m/Λ becomes of order 1 or even larger and hence a series inm/Λ does not converge anymore and the EFT breaks down. In terms of the photon examplethis means that the effective theory breaks down if the photon wavelength becomes of the orderof the lattice spacing in the crystal since Bragg-scattering cannot be neglected anymore. Thus,
8
one has to ensure that the expansion parameter of an EFT is indeed small in the energy regionwhere one wants to investigate the original theory.
1.3.1 EFT’s for QCD
After this rather general introduction to EFT’s two important effective theories for low-energyQCD are presented.
Chiral perturbation theory
For energies up to few GeV where the interaction of nucleons (or other heavier hadrons) canbe described by pion exchange (or if one includes the strange quark, also by kaon or eta-mesonexchange) a famous EFT called chiral perturbation theory (ChPT) based on the work of Weinbergin Ref. [65] and of Gasser and Leutwyler in Refs. [66, 67] exists. This theory uses the factthat the masses of the two (or, including s, three) lightest quarks are relatively small (seeEq. (1.1)). Hence, one can impose chiral symmetry which states that in a vector gauge theorywith massless fermions, the right- and left-handed components of the latter can be transformedindependently. Due to the existence of the quark condensate and the pion decay constant fπ 6=0 the chiral symmetry is spontaneously broken with the three pions being the correspondingGoldstone bosons. To be more precisely the pions are pseudo-Goldstone bosons since the chiralsymmetry is only an approximate symmetry because u and d quarks are not massless in nature.In fact, the mass of the quarks is also the reason why one cannot construct a ChPT includingthe very massive charm, bottom or top quarks. A extensive overview of ChPT can be found inRef. [68].
Pionless effective field theory
As mentioned above ChPT is valid up to a few GeV, but especially in nuclear physics the energiesare often even smaller, i.e. in the range of a few MeV. The question arises if one can constructan effective field theory also in this energy region. Indeed, this is possible: for energies where thepion exchange between heavy hadrons like nucleons is not relevant one can treat these hadronsas non-relativistic point-like particles which only interact via contact interactions. Such a theorywith an expansion parameter Q/mπ with Q being the internal momentum is called pionlesseffective field theory (EFT(/π)). This was introduced in Refs. [69, 71–73]. It was derived fornucleon–nucleon interactions and is a commonly used tool in nuclear physics [74–77]. Especiallythe deuteron was very successfully analyzed in EFT(/π) (see Refs. [71–73, 78, 79, 93]). Since onecan treat the deuteron as a simple hadronic molecule made of two nucleons, one concludes thatEFT(/π) might also be suitable for the description of other hadronic molecules as long as theirconstituents have masses of the order of 1 GeV or more and their binding momenta (whichdefine the internal momenta) are much smaller than the pion mass. However, also for lighter(but still reasonably heavier than pions) constituents or binding momenta around mπ one coulduse a pionless effective field theory to obtain at least some first insights into such a system.A comprehensive review of EFT(/π) and the universality in few-body physics can be found inRef. [80].
9
1.4 Basic scattering theory
In this section we derive all relations of the effective range expansion (ERE) in scattering theoryup to second order (next-to leading order, NLO) for the sake of completeness. However, through-out this thesis we will only consider the leading order (LO) effective range expansion.We start with quantum mechanics and follow Refs. [81,82]: for a finite range potential V whichfalls off sufficiently fast for large r the wave function of a scattered particle has the followingasymptotic form:
ψ(r) = eiki·r + f(k, θ)eikf r
r, (1.6)
where the scattering amplitude f(k, θ) depends on the modulus k := |ki| = |kf | of the incoming(index i) and outgoing momenta (index f) and on the angle θ between those, i.e. k2 cos θ = ki ·kf .If one considers a central potential V (r) ≡ V (r) one can expand the scattering amplitude inpartial waves with angular momentum L. Ignoring additional spin for the moment the expansionhas the form [81,82]:
f(k, θ) =1
k
∑
L
(2L+ 1)eiδL sin(δL)PL(cos θ) , (1.7)
with the Legendre polynomials PL and a phase shift δL. As we will deal with S-wave states inthe following chapters we will now assume L = 0 and conclude from Eq. (1.7) that the scatteringamplitude f is independent of θ and thus only a function of k. Since S-wave scattering isspherical symmetric one can furthermore integrate the incoming plane wave in Eq. (1.6) over thesolid angle to find
∫dΩ
4πeiki·r =
sin(kr)
kr, (1.8)
which can be written as
sin(kr)
kr= − 1
2i
e−ikr
kr+
1
2i
eikr
kr. (1.9)
As only the outgoing wave eikr can be affected by the scattering process and since the particlenumber is conserved, the only possible modification due to the potential is a phase shift δ ≡ δL=0
in the outgoing wave, that is,
eikrscattering−→ ei(kr+2δ) = eikr + 2ieikreiδ sin δ . (1.10)
Hence, the asymptotic wave function in case of S-wave scattering has the form:
ψ = eiki·r +eiδ sin δ
k
eikf r
r, (1.11)
which has to be compared with Eq. (1.6) in order to find a relation for the scattering amplitudein terms of the phase shift:
f(k) =eiδ sin δ
k=
1
k
sin δ
cos δ − i sin δ=
1
k cot δ − ik . (1.12)
10
Bethe has shown in Refs. [83, 84] that for a finite range potential one can expand k cot δ aboutk = 0 yielding up to second order
k cot δ = −1
a+
1
2r0k
2 + ... , (1.13)
where the scattering length a and the effective range r0 were introduced (the reason why thisapproximation is called effective range expansion). Next, one can use Eq. (1.13) and Eq. (1.12)to finally get the second order ERE S-wave scattering amplitude:
fERENLO(k) =1
− 1a
+ 12r0k2 − ik . (1.14)
In the next step we identify a relation between the quantum mechanical scattering amplitudef and the interaction part T of the scattering matrix S = 1 + T in quantum field theory. Forthis purpose we again consider the wave function ψ of Eq. (1.6) which obeys the Schrodingerequation
(H0 − E)ψ = −V ψ , (1.15)
where H0 is the free Hamilton operator. Inverting this equation one finds that for 1/(H0 − E)being the inverse of the operator H0 − E it holds:
ψ = − 1
H0 − EV ψ = φ− 1
H0 − EV ψ , (1.16)
with the free particle solution φ = eiki·r which fulfills (H0 −E)φ = 0. However, a particle can ingeneral undergo multiple scattering processes. In order to take this fact into account one has toextend Eq. (1.16) using the T -matrix:
ψ = φ− 1
H0 − ETφ . (1.17)
The derivation of this relation can be found for example in Ref. [1] where it is shown that onecan schematically define the T -matrix in the following way:
T := V + V1
H0 − EV + V
1
H0 − EV
1
H0 − EV + ... = V + V
1
H0 − ET . (1.18)
With this knowledge one can show that the quantum mechanical S-wave scattering amplitude fis related to the QFT T -matrix by
T (k) =2π
µf(k) , (1.19)
where the prefactor 2π/µ (with reduced mass µ) accounts for the different normalization in QFT.According to Eq. (1.19) the T -matrix itself is often called scattering amplitude and we will doso, too. Using what we have found in Eq. (1.14) one obtains the following first and second orderERE S-wave scattering amplitude (or T -matrix):
TERELO (k) = −2π
µ
11a
+ ik(1.20)
TERENLO (k) = −2π
µ
11a− 1
2r0k2 + ik
, (1.21)
from which, however, we need in the following chapters only the LO relation.
11
Im k
Re k
Figure 1.4: Poles of the S-matrix in the complex momentum plane of k. Bound states corres-pond to poles marked by a dot on the positive imaginary axis, virtual states to the ones markedby diamonds on the negative imaginary axis. The remaining crosses and squares indicate poleson the second sheet, i.e. in the lower half of the k-plane. They are either identified as resonances(being states with definite quantum numbers so that one can interpret them as particles) if theyare close to the positive real axis (crosses) or else as non-resonant background scattering effects(which could not be interpreted as particles) if they are far away from the positive real axis(squares).
1.4.1 Some remarks on resonances, bound and virtual states
The scattering of two particles might be affected by inelastic effects due to intermediate two-bodystates which are either resonances, bound or virtual states. These states correspond to poles inthe S-matrix (or equivalently in the T -matrix since S = 1+T ) and are classified by the positionof the respective pole in the complex momentum plane as it is for example explained in Ref. [82].A bound state of two particles with masses m1 and m2 (and reduced mass µ) corresponds to apole at k = iγ with γ > 0, namely, to a pole with vanishing real part, but non-zero imaginarypart. The energy of this state is EB = k2/(2µ) = −γ2/(2µ) < 0 which is identified with the(negative) binding energy. Moreover, this relation also motivates the term binding momentumfor γ. Note, that we will define (using the bound state mass M12) a quantity B := m1 +m2−M12
which has positive values for bound states so that B = −EB and call B as well ”binding energy“.Using B instead of EB one can write γ as γ =
√−2µEB =√
2µB which will be repeatedly usedin this work. For energies below its threshold m1 + m2 + EB = m1 + m2 − B a bound stateis stable according to the force which has generated it (which is the strong force for hadron–hadron scattering). However, there could be other forces in nature which can cause a decay of abound state also below threshold via the decay of its constituents (in our case of hadrons theseforces would be the electromagnetic or weak force). Next, we consider virtual states which arein some sense the counterparts of bound states with the difference that they are located on the
12
second sheet, i.e. in the lower half of the k-plane. They lie on the negative imaginary axis againcorresponding to a pole at k = iγ, but with γ < 0. Their energy EB = k2/(2µ) = −γ2/(2µ) < 0is the same as for bound states. However, the quantity B = m1 + m2 − M12 is negative forvirtual states which thus are sometimes called anti-bound. Since the sum of the constituentmasses (m1 +m2) is smaller than the mass of the two-body state (M12), virtual states cannot beobserved in nature. But due to the corresponding pole in the S-matrix they affect the scatteringof the two particles. The ”binding momentum“ γ of virtual states is negative, but also the bindingenergy B is smaller than zero. Hence, one must modify the relation between them according to
γ = sgn(B)√
2µ|B| , (1.22)
which is valid for both bound and virtual states. Finally, resonances are dynamically generatedstates located on the second sheet somewhere below, but close to the positive real axis. Theythus have a rather small imaginary part (−ki, ki > 0) of the momentum, but also a non-vanishingreal part (kr) corresponding to a kinetic energy which is the reason for their widths as it can beseen from
E =k2
2µ=
(kr − iki)2
2µ=k2r − k2
i − 2ikrki2µ
:= Er − iΓ
2, (1.23)
where Er is the resonance position and Γ the resonance width. From Eq. (1.23) it is also clearthat the widths grows as the imaginary part of k becomes larger. Hence, a resonance far awayfrom the real axis (i.e. with a large imaginary part ki) is so broad that it cannot be interpretedas particle anymore, but rather as some non-resonant background in the scattering process.In Fig. 1.4 we have summarized the classification of resonances, bound and virtual states bysketching the corresponding poles of the S-matrix in the complex momentum plane of k.In general hadronic molecules can be either bound or virtual states or resonances since all ofthem can be considered as particles with definite quantum numbers. However, we will below usethe binding momentum as variable in the scattering amplitudes of processes including hadronicmolecules. Hence, the particles we consider as molecular states should be either bound or virtualstates where γ is well-defined. Nevertheless, one can – motivated by the often large experimentalerrors in the masses – argue that one could also assign to resonances which are located not toofar away from the imaginary axis at least an approximate binding momentum.
1.4.2 Pole position and effective range expansion
In the first part of this section the ERE amplitude was derived. Now we will relate the pole posi-tion k = iγ of bound states to this expansion. Considering Eq. (1.14) or equivalently Eq. (1.20)one concludes that the scattering amplitude f(k) or T (k) has a pole for a vanishing denominator.At LO this leads to
−1
a− ik = −1
a+ γ
!= 0 , (1.24)
which yields
γ =1
a. (1.25)
13
At NLO the denominator of the scattering amplitude is a function of k2 and hence one finds
−1
a+
1
2r0k
2 − ik = −1
a− 1
2r0γ
2 + γ!
= 0 . (1.26)
This quadratic equation has two solutions
γ1,2 =1
r0
± 1
r0
√1− 2
r0
a. (1.27)
For a large scattering length a r0 being much larger than the effective range r0 one can expandthe square root in Eq. (1.27) using
√1− x ≈ 1− x/2 +O(x2) (valid for small x) to obtain
γ1,2 =1
r0
± 1
r0
(1− r0
a
)=
2r0− 1
a
ar0≈ 2r0
1a
, (1.28)
where γ1 ≈ 2/r0 corresponds to a deeply bound pole and where γ2 = 1/a reproduces the resultEq. (1.25) found at LO which corresponds to a shallow bound state with energy EB = γ2
2/(2µ) =1/(2µa2) which only depends on the scattering length.Note, that the same derivation can be done for virtual states with a pole also at k = iγ, but withγ < 0 and hence one concludes from Eq. (1.25) that virtual states have a negative scatteringlength.
14
Chapter 2
Universality and the Efimov effect
So far we have only considered two-body systems. In the following we move on to three-bodyphysics, in particular we discuss the phenomenon of the emergence of a three-body bound statespectrum in systems with large scattering length. This effect is an example of the universal scalingbehavior of systems with large scattering length (universality) and known as Efimov effect. Itwas proposed by Efimov [85], theoretically proven in Ref. [86, 87] and experimentally observed,first in Ref. [100] (which is also reviewed in Ref. [88]). In a system of three identical spin- andisospinless bosons with divergent scattering length the Efimov effect leads to an infinite numberof three-body bound states (trimers) whose binding energies B3 are geometrically spaced andwhich accumulate at the three-body scattering threshold. The spacing is defined by the so-calledscaling factor :
B(n+1)3
B(n)3
≈ 515.03 . (2.1)
Naively, one expects that in a scattering process the parameters of the ERE are all of the sameorder which is defined by the interaction range R of the corresponding potential V (r) → 0 forr > R. Up to second order (i.e. NLO) this would mean that the S-wave scattering length a andeffective range r0 would scale as R. However, there could be systems where the scattering lengthis large, that is, much larger than the interaction range a R and hence much larger than theeffective range a r0 ∼ R. Such systems with a large scattering length a→∞ (i.e. systems inthe resonant limit) exhibit a so-called universal scaling behavior, meaning that observables canbe given in terms of the scattering length only. Besides in system where the scattering lengtha r0 ∼ R itself is large one finds an universal behavior also in systems where the interactionrange goes to zero (scaling or zero-range limit).Although there are no systems with divergent scattering length in nature, there are systemswith reasonably large a. For example the nucleon–nucleon 1S0 system with scattering lengthaNN = −18.7(6) fm [89] which is approximately an order of magnitude larger than the effectiverange r0,NN = 2.75(11) fm [90] in this channel. Moreover, systems treated in pionless EFTonly interact via contact interactions (see section 1.3) which means that the interaction range iszero. Therefore every system that can be described in EFT(/π) is universal up to higher ordercorrections. Hence, the scattering of two particles with large, but finite scattering length has anuniversal scaling behavior as long as it is allowed (γ < mπ) to analyze it in EFT(/π) where the
15
interaction range is zero. Consequently, an accordant three particle system could be affected bythe Efimov effect with an exact scaling behavior like that in Eq. (2.1). However, although thenumber is constant for all n it must not necessarily be 515.03. A further difference for systemswith finite scattering length is that the trimer spectrum is cut off at the two-body (dimer)threshold B2. Hence, there are not infinitely many Efimov trimers, but only as much as arefitting into the energy region B
(0)3 < E < B2.
The Efimov effect and universality in few-body systems in general are discussed in Ref. [80] ofHammer and Braaten which in particular also contains a review of many important methods inEFT(/π) like the concept of dimer auxiliary fields. The following summary is based on this work.
2.1 Efimov effect for identical bosons
We consider a system of three identical particles ψ of mass m, but without spin or isospin degreesof freedom. Additionally, we assume that they have a two-body interaction with large scatteringlength and a three-body interaction. In the scaling limit, namely, for a vanishing interactionrange, such a system can be described by the non-relativistic Lagrangian density [80]
L = ψ†(i∂t +
∇2
2m
)ψ − g2
(ψ†ψ
)2 − g3
(ψ†ψ
)3. (2.2)
Here, we use the so-called dimer field trick [91, 92] instead of the Lagrangian introduced inEq. (2.2). Therefore we introduce an auxiliary dimer field d which represents a two-body boundstate of two ψ particles. With this new field one can write down a new Lagrangian density,
Ld = ψ†(i∂t +
∇2
2m
)ψ + g2 d
†d− g2
(d†ψ2 +
(ψ†)2d)− g3 d
†dψ†ψ , (2.3)
which is equivalent to Eq. (2.2). This equivalence can be seen by eliminating the d field fromEq. (2.3) using its equation of motion as it is shown in Ref. [80]. We note, that there is nokinetic term for the dimer field (which would contain time derivatives) and thus it is not dynamic.Nevertheless, it interacts with the field ψ via the last two terms in Eq. (2.3). The first interactionterm proportional to g2 thereby describes the decay of the dimer field into two boson fields ψ.Thus, it leads to self-energy corrections to the constant bare propagator i/g2 of the field d. Atthe end, these corrections allow the dimer field to propagate in space and time. To show thisexplicitly we consider the Dyson equation shown in Fig. 2.1(a) as an infinite series of Feynmandiagrams where a thick solid line represents the bare and a double line the full propagator of thedimer field.Naming the full one as iD(p0,p) and the bare one iD0 = i/g2 one can – according to Fig. 2.1(a)– write down a relation for the full propagator:
iD(p0,p) = iD0 + iD0 iΣ iD0 + iD0 iΣ iD0 iΣ iD0 + ...
= iD0
[ ∞∑
n=0
(−ΣD0
)n]
=iD0
1− (−ΣD0)=
i
(D0)−1 + Σ, (2.4)
16
= + + + ...
(a)
iΣ(p0,p) =
(b)
Figure 2.1: Dyson equation for the dimer field in the three identical boson system as infiniteseries of boson loops. A thick solid line represents the bare and a double line the full dimerfield propagator while a single line stands for the ψ propagator (a). Self-energy diagram for theidentical boson system. The single lines in the loop represent the boson fields ψ (b).
where we have introduced the self-energy Σ and used the geometric series to simplify the infinitesum. In order to find an expression for the self-energy we firstly identify the correspondingFeynman diagram (Fig. 2.1(b)) and then use the Feynman rules following from the Lagrangianin Eq. (2.3) to obtain
Σ(p0,p) = 2g22
∫d4q
(2π)4
1
p0 + q0 − 12m
(p2
+ q)2
+ iε
1
−q0 − 12m
(p2− q
)2+ iε
, (2.5)
with the loop momentum (q0,q). We have used that the non-relativistic propagator of the fieldψ is given by
iS(p0,p) =i
p0 − |p|2
2m+ iε
. (2.6)
Note, that the extra factor of 2 in Eq. (2.5) is a symmetry factor due to the fact that the bosonsare identical. Applying the residue theorem one can carry out the integral over dq0 yielding
Σ(p0,p) = 2g22
∫d3q
(2π)3
i m
q2 −mp0 + p2
4− iε
, (2.7)
which can now be calculated in dimensional regularization with a scale µ using the relation
∫ddq
1
(q2 + 2 q · k − b2)α= (−1)
d2 iπ
d2
Γ(α− d
2
)
Γ(α)
[−k2 − b2
] d2−α
, (2.8)
which can be found in many QFT textbooks, for example in Ref. [1]. In D = 4 dimensions (i.e.d = D − 1), with k = 0 and Γ(−1/2) = −2
√π we thus end up with:
Σ(p0,p) = limD→4
(2g2
2 µD−4
∫dD−1q
(2π)D−1
i m
q2 −mp0 + p2
4− iε
)= −mg
22
2π
√−mp0 +
p2
4− iε . (2.9)
17
i Tel =
Figure 2.2: Elastic scattering amplitude of two identical bosons ψ with an intermediate dimerstate d.
Hence, the full propagator of the dimer field can according to Eq. (2.4) be written as
iD(p0,p) =i
g2 − mg222π
√−mp0 + p2
4− iε
= − 2π
mg22
i
− 2πmg2
+√−mp0 + p2
4− iε
. (2.10)
In the next step we relate the full propagator to the binding momentum γ of the dimer. Hence,we consider the elastic scattering of two boson with an intermediate dimer state (Fig. 2.2). Thecorresponding amplitude is given by
Tel(k) = −4g22D
(p0 = E =
k2
m,p = 0
)=
8π
m
1
− 2πmg2
+√−k2 − iε
= −8π
m
12πmg2
+ ik, (2.11)
with k being the modulus of the incoming and outgoing momenta which are equal in an elasticprocess. Additionally, we have taken into account a symmetry factor of 4 due to the identicalbosons. Comparing this result with the leading order ERE amplitude in Eq. (1.20) we deducethat the scattering length and thus the binding momentum (cf. LO version of Eq. (1.25)) canbe identified as
1
a≡ γ =
2π
mg2
. (2.12)
This allows us to finally write the full propagator of the dimer field as
iD(p0,p) = − 2π
mg22
i
−γ +√−mp0 + p2
4− iε
. (2.13)
Moreover, this leads to a wave function renormalization constant Z defined as the residue of thepole in the full propagator [1] and given by
Z =4π γ
g22m
2. (2.14)
2.1.1 Three-body scattering amplitude
The considerations above provide in some sense a tool box for the following. Namely, the deter-mination of the three-body scattering amplitude, known as Skorniakov–Ter-Martirosian (STM)equation [94, 95]. Firstly, we note that this task is reasonably simpler using an auxiliary dimerfield instead of using the Lagrangian in Eq. (2.2). Still following Ref. [80] we have to solve the
18
= + ++
Figure 2.3: Integral equation for the three-body scattering of identical bosons using the dimerfield trick. A black square represent a two-body dimer–boson vertex and the black dot the three-body vertex.
integral equation shown in Fig. 2.3 in terms of Feynman diagrams. This representation can beunderstood in the sense that the repeatedly insertion of the right-hand-side into itself will gen-erate all allowed diagrams with an arbitrary number of bosons. The corresponding scatteringamplitude is – using the Feynman rules according to the Lagrangian density Eq. (2.2) – in thecenter-of-mass system given by
t(E,k,p) = −[
4g22
E − k2
2m− p2
2m− (k+p)2
2m+ iε
+ g3
]
+ i
∫d4q
(2π)4t(E,k,q)
D(E + q0,q)
−q0 − q2
2m+ iε
[4g2
2
E + q0 − p2
2m− (p+q)2
2m+ iε
+ g3
], (2.15)
with incoming 4-momentum k, outgoing one p, the center-of-mass energy E and a symmetryfactor of 4 in front of g2
2. Applying the residue theorem to solve the dq0 integration sets q0 =−q2/(2m). After multiplying with the wave function renormalization derived in Eq. (2.14), oneends up with the following equation for the renormalized amplitude T := Z t:
T (E,k,p) = − Z[
4g22
E − k2
2m− p2
2m− (k+p)2
2m+ iε
+ g3
]
+
∫d3q
(2π)3T (E,k,q)D
(E − q2
2m,q
)[4g2
2
E − q2
2m− p2
2m− (p+q)2
2m+ iε
+ g3
].
(2.16)
We will now restrict ourselves to the analysis of the S-wave three-body system. As explained inappendix D one can project out the L = 0 partial wave by applying the projection operator
1
2
∫ 1
−1
d cos(θ) PL(cos θ) .
The remaining integral over d3q can be rewritten in spherical coordinates and hence one ends up –after carrying out the integrals over the angles – with an expression proportional to the Legendrefunction of the second kind QL (see appendix D). In Eq. (D.15) it was derived a representationof QL=0 in terms of the logarithm. Hence, the three-body scattering amplitude in Eq. (2.16) only
19
depends on the moduli of the momenta and reads:
T (E, k, p) =16π γ
m
[1
2kpln
(p2 + pk + k2 − E − iεp2 − pk + k2 − E − iε
)− 1
4m
g3
g22
]
+4
π
∫ Λ
0
dqq2 T (E, k, q)
−γ +√−mE + 3
4q2 − iε
[1
2qpln
(p2 + pq + q2 − E − iεp2 − pq + q2 − E − iε
)− 1
4m
g3
g22
],
(2.17)
where the remaining integral over dq is regularized by a cutoff Λ. To account for a correctrenormalization it is now convenient to introduce a cutoff dependent coupling constant H(Λ) [80],
g3 = −4mg22
Λ2H(Λ) , (2.18)
which yields
T (E, k, p) =16π γ
m
[1
2kpln
(p2 + pk + k2 − E − iεp2 − pk + k2 − E − iε
)+H(Λ)
Λ2
]
+4
π
∫ Λ
0
dqq2 T (E, k, q)
−γ +√−mE + 3
4q2 − iε
[1
2qpln
(p2 + pq + q2 − E − iεp2 − pq + q2 − E − iε
)+H(Λ)
Λ2
].
(2.19)
Note, that the equation above is equivalent to Eq. (336) in Ref. [80] up to the fact that the massof the bosons is not set to m = 1. As discussed in Ref. [80] the function H(Λ) must compensateevery change in the cutoff Λ so that the amplitude itself is well-behaved in the limit of Λ goingto infinity. However, H(Λ) is proportional to the three-body coupling g3. Hence, one concludesthat in a system without a three-body interaction it holds H = 0 and the divergent part of theamplitude drops out. In the other case where g3 6= 0 one thus needs a three-body observable inorder to find the physical value of H(Λ) which ensures the right behavior of the amplitude in thelimit Λ→∞. In fact, a three-body bound state with binding energy E3 = −B3 corresponds toa pole in Eq. (2.19) exactly at the energy E = −B3. Thus, one can fix the cutoff dependence ofH(Λ) by determining the poles in Eq. (2.19) for a varying cutoff and searching for the right poleposition E = −B3. If this is done one can calculate for every cutoff Λ the corresponding valueof H. In Ref. [80] it is shown that a numerical determination of H yields
H(Λ) =cos[s0 ln
(ΛΛ∗
)+ arctan (s0)
]
cos[s0 ln
(ΛΛ∗
)− arctan (s0)
] , (2.20)
which depends on two parameters: on the one hand Λ∗ which must be fixed using a three-bodyobservable and on the other hand the scaling parameter s0 = 1.00624 which is independent ofthe three-body interaction and which can be determined as discussed in the next subsection.
20
Transcendental equation for the scaling parameter
As mentioned previously the scaling parameter s0 = 1.00624 is not affected by any three-bodyphysics. Therefore one can assume for simplicity that g3 = 0 and hence, that H(Λ) vanishes(according to the fact that there always exists a cutoff Λ so that H(Λ) = 0). In Eq. (2.19) onecan then take the limit of Λ→∞ and one finds
T (E, k, p) =16π γ
m
1
2kpln
(p2 + pk + k2 − E − iεp2 − pk + k2 − E − iε
)
+4
π
∫ ∞
0
dqq2 T (E, k, q)
−γ +√−mE + 3
4q2 − iε
1
2qpln
(p2 + pq + q2 − E − iεp2 − pq + q2 − E − iε
). (2.21)
This equation is scale invariant and symmetric under the change q → 1/q (”inversion invariant“).In Ref. [80] it is argued that this invariance causes that in the limit of asymptotic large outgoingmomenta p the amplitude T (E, k, p) has a solution in form of a power law ps. Furthermore,the authors motivate that in the limit p → ∞ one can neglect the inhomogeneous term and inaddition all variables E and γ which are proportional to the incoming momentum k p→∞.Consequently, one ends up with
T (p) =4√3π
∫ ∞
0
dq
pT (q) ln
(p2 + pq + q2
p2 − pq + q2
). (2.22)
Redefining the amplitude via T (p) = pT (p),
T (p) =4√3π
∫ ∞
0
dq
qT (q) ln
(p2 + pq + q2
p2 − pq + q2
), (2.23)
and inserting the power law solution T (p) ∼ ps yields in the asymptotic momentum limit
ps =4√3π
∫ ∞
0
dq qs−1 ln
(p2 + pq + q2
p2 − pq + q2
). (2.24)
Following Ref. [80] it is useful to substitute X = q/p so that Eq. (2.24) finally simplifies to therelation
1 =4√3π
∫ ∞
0
dX Xs−1 ln
(X + 1
X+ 1
X + 1X− 1
). (2.25)
The remaining integral is related to a Mellin transform (see e.g. Ref. [168]) and can be analyticallysolved. Hence, one obtains a transcendental equation for the scaling parameter s [80]:
1 =8√3π
1
s
sin(π6s)
cos(π2s) . (2.26)
This equation has a purely imaginary solution s = is0 with s0 = 1.00624. Thus, Eq. (2.26) is therelation which defines the scaling parameter s0 in the three-body problem. As a remark, note
21
that one could have found Eq. (2.26) also by rewriting the logarithm in terms of the Legendrefunction of the second kind using Eq. (D.15),
ln
(X + 1
X+ 1
X + 1X− 1
)= 2Q0
(X +
1
X
),
and considering the results discussed in appendix E.The purely imaginary solution of Eq. (2.26) tells us that the amplitude T (p) ∼ ps has not oneuniquely determined real solution, but instead two linearly independent complex solutions (seealso Ref. [167]). Consequently, there is a discrete scaling invariance, that is, a discrete scalingfactor exp(π/s0) in the system of three identical bosons. This discrete scaling invariance ismanifestly an effect of Efimov physics [80] and one concludes that a different system of threeparticles which has not a purely imaginary exponent in the power law solution will not be affectedby the Efimov effect, meaning that there is no three-body bound state (the Efimov trimer) insuch a system. It is therefore important to note on the one hand that one does not need anexplicit three-body force in the Lagrangian density in order to check if a three particle system isaffected by the Efimov effect. On the other hand Eq. (2.26) is not an universal relation; it wasderived for a system of three identical, spin- and isospinless bosons, but it will most probablychange if one or more of these properties are changed. These facts motivate the following work:we will derive an analogous transcendental equation depending on some parameters in orderto describe a large number of different three particle systems (including spin, isospin, identicalor distinguishable bosons and fermions, etc.). Such an equation allows to simply check for an(almost) arbitrary three particle system if the scaling parameter has a purely imaginary solution,that is, to check if the Efimov effect is present in the considered system. Moreover, this canbe done without any knowledge of possible three-body physics (corresponding to the availableexperimental data of most hadronic molecule candidates) since in the above derivation we haveset the three-body coupling constant H equal to zero anyway. Only to obtain cutoff independentthree-body observables would require to fix H(Λ) to its physical value.
2.2 Efimov physics in cold atoms and halo nuclei
Up to now the discussion of the Efimov effect as an universality phenomenon was restricted tonuclear physics and – where this work focuses on – to hadronic molecules, i.e. particle physics.Besides these applications the Efimov effect is also an important phenomenon in cold atoms andhalo nuclei. The former are atoms which are trapped (e.g. in a magneto-optical trap) and cooleddown by different techniques like laser cooling until they reach temperatures very close to zeroKelvin (for an overview of this topic see for example Ref. [96]). At such low temperatures – whichare equivalent to low energies – quantum effects become important and thus cold atoms are aperfect testing ground for phenomena like Bose-Einstein condensation, but in fact also for theEfimov effect. Especially, the existence of Feshbach resonances [97–99] which allow to fine-tunethe scattering length of a two-atom state to be ”large“, leads to many experimental observationsof the Efimov effect. In bosonic systems it was found using cesium [100,101], potassium [102] orlithium [103, 104] isotopes, but also in a fermionic system of a different lithium isotope Efimovphysics were observed [105,106]. The mentioned experiments have in common that they use the
22
indirect detection method of an enhanced recombination rate [109–111], but there are also directobservations of Efimov trimers [107,108].The second field where universality effects are important are halo nuclei. The idea behind thisterm is that some isotopes of a few different elements can be interpreted as a compact, deeplybound core and one or more orbiting nucleons (halo-nucleons) [112–115]. This picture has ledto the development of the so-called Halo EFT where one treats the core and the halo-nucleonsas shallowly bound few-body system. The expansion parameter is R/a with R being the rangeof the core–halo-nucleon interaction and its scattering length a [116, 117]. Both, core and halo-nucleons are described as non-relativistic fields and hence the degrees of freedom in the theory aredrastically reduced (for more details on Halo EFT see the reviews in Refs. [118,119]). Examplesfor halo nuclei are – besides others – 11Li with two surrounding neutrons, 11Be with one neutronor 8B with a halo-proton [120]. Thus, one concludes that there are in particular also three-bodysystems made of a core and two nucleons. Since the core–nucleon two-body scattering length ais expected to be large compared to the range R of the core–nucleon interaction, it is likely thatsuch a system might be affected by the Efimov effect. Indeed, there was much effort to quantifythis assumption about the Efimov effect in halo nuclei [121–127] (see the review in Ref. [128]).The final conclusion of this subsection is that Efimov physics are an important part of manydifferent fields in physics which – at least on the first sight – deal with very different subjects.Already at this point we want to emphasize that the considerations in the following chapters arenot restricted to hadronic molecules and in particular that the final transcendental equation forthe scaling parameter (which tells us if the Efimov effect exists in a system) can also be appliedto a system of cold atoms or to a three-body halo nucleus.
23
Chapter 3
Efimov effect in a general three particlesystem
In section 2.1 we have explained how the Efimov effect occurs in a system of three identical spin-and isospinless bosons. At the end we showed a transcendental equation for the exponent s ofthe power law solution in the limit of asymptotic large momenta. In case of S-wave scatteringwe have found:
1 =8√3
1
s
sin(π6s)
cos(π2s) . (3.1)
From the derivation of this equation we know that its structure depends on the following particleproperties: species (bosons, fermions or mixture), relation of masses as well as angular momen-tum, spin and isospin quantum numbers of the dimer and its constituents. The question is now:how will this equation change if one changes one or more of these properties? To answer thisquestion we will derive a general transcendental equation with a number of parameters represen-ting various three particle configurations. However, choosing one specific three particle systemwith known dimer states the parameters straightforwardly reduce to numbers and at the end onesimply has to find the eigenvalues of a real matrix.
3.1 General dimer states
Let us consider three isospin multiplets A1, A2, A3 and their corresponding anti-multiplets A1,A2, A3 which are related via the G-parity operator G,
Ai = GAi = CeiπI2Ai , (3.2)
where C is the charge conjugation operator and eiπI2 represents a rotation in isospin space aroundthe I2 axis, that is, a change from I3 = ±1/2 to I3 = ∓1/2. Because Ai = GAi their masses areequal and hence there are only three mass parameters m1, m2 and m3. Nevertheless, we treatAi and Ai as independent states since we work in a non-relativistic theory.
24
A general system of three particles P1, P2, P3 can contain both particles and anti-particles. Wedefine these particles is terms of the multiplets introduced above:
P1 := a1A1 + b1A1
P2 := a2A2 + b2A2
P3 := a3A3 + b3A3 , (3.3)
with ai = 0, 1 and bi = 1, 0 ∀ i ∈ 1, 2, 3. Hence, choosing e.g. the system P1 = N , P2 = N ,P3 = Λ this would fix the parameters to be b1 = a2 = a3 = 1 and a1 = b2 = b3 = 0.We assume that these particles have shallow two-body bound or virtual states dij which we willfor simplicity both call ”dimers“ (and which in general are isospin multiplets as well). Sincethe dimer d12 between particle P1 and P2 is physically identical to the dimer d21 between P2
and P1 there only are three different dimers in the system. Note, that mathematically dij 6= djidue to possible fermion minus signs from interchanging Pi and Pj (physically such a minus signcan be absorbed into the coupling constant which does not affect the observables). Thus, onehas to choose one convention for the order of appearance of Pi and Pj, but which of them isirrelevant since the observables are independent of this choice (of course if one does not mix bothconventions). We will use the convention
dij ∼ PjPi with i < j ∈ 1, 2, 3 , (3.4)
and the three possible dimers are thus given as d12, d13 and d23.
3.1.1 Dimer flavor wave function
The possible flavor wave functions for a dimer dij are
dij = AjAi (3.5)
dij = AjAi (3.6)
dij =
1√2
(AjAi + ηijAjAi
), if baryon number and flavor are 0 and Ai 6= Aj
ηijAjAi , if baryon number and flavor are 0 and Ai = Aj
AjAi , else
. (3.7)
In the third equation above the dimer has well-defined G-parity with G-parity quantum numberηij = ±1 if the baryon number and the flavor meaning strangeness, charm, beauty (bottomness)and – for completeness although hadrons containing top quarks does not exist due to the shortlifetime of t quarks – topness are all equal to zero. This condition implies that for Ai 6= Aj (e.g.B∗B) the combinations AjAi and AjAi have the same quark content with the same quantumnumbers and masses. Thus, a possible dimer must be a superposition of both states. The G-parity quantum number ηij in Eq. (3.7) fixes the sign within the superposition in the followingmanner: the G-parity eigenstate dij must fulfill Gdij = ηijdij and since quantum mechanics tellus that there could be a phase eiϕ between the two states of the superposition, we start with thegeneral wave function
dij =1√2
(AjAi + eiϕAjAi
)=
1√2
(AjAi + eiϕG
(AjAi
)), (3.8)
25
and apply G to both sides:
Gdij = G 1√2
(AjAi + eiϕG
(AjAi
))=
1√2
(GAjAi + eiϕ
(AjAi
))
!= ηijdij =
1√2
(ηijAjAi + ηije
iϕG(AjAi
)). (3.9)
Comparing the coefficients one finds that for ηij = ±1 the phase factor must be eiϕ = ±1. There-fore one can replace in Eq. (3.8) the phase factor by the G-parity quantum number which leadsto the wave function in Eq. (3.7). In the case with Ai = Aj (e.g. B∗B∗) the prefactor ηij doesnot affect observables since later on it could be absorbed in the coupling constant. However, forthe sake of consistency and to distinguish this case from the ”else“ case where G-parity is not agood quantum number we will keep it.Using ai and bi one can parametrize the flavor wave function in a way that for a given system P1,P2, P3 (which fixes all ai’s and bi’s to be either 0 or 1) the correct wave function is automaticallygenerated. For this we introduce two short-hand notations. Firstly, we need a parameter whichtells us if the considered dimer has a well-defined G-parity. This is the case if the following quan-tum numbers of the dimer are zero: baryon number, strangeness, charm, bottomness (beauty)and topness must vanish because only hadrons consisting either solely of up and down quarks oran arbitrary number of qq pairs of heavier quarks are up to a sign invariant under the applicationof the G-parity operator G. In order to avoid the lengthy combination of Kronecker-deltas whichensure the vanishing of all the quantum numbers we instead define:
δη|1| := δbaryon number 0 × δstrangeness 0 × δcharm 0 × δbeauty 0 × δtopness 0 , (3.10)
motivated by the fact that if the right-hand-side is 1, i.e. if all mentioned quantum numbersindeed vanish, then the dimer is a G-parity eigenstate with corresponding quantum numberη = ±1 = |1|. The second short-hand notation is:
δAiAj =
1 , if Ai is identical to Aj, i.e. Ai = Aj
0 , else. (3.11)
Note here, that
δPiPj = 1 ⇒ δAiAj = 1 ,
δAiAj = 1 ; δPiPj = 1 . (3.12)
With this notation one can parametrize the dimer flavor wave function as follows:
dij = aiaj AjAi + bibj AjAi
+ δηij |1|
[1√2
+ δAiAj
(1− 1√
2
)](biaj + aibj)
[(ηij)
δAiAj AjAi +(1− δAiAj
)ηijAjAi
]
+(1− δηij |1|
)biajAjAi ∀ i < j ∈ 1, 2, 3 . (3.13)
To clarify this notation let us consider the three particle system B∗BB∗. Consequently, wehave P1 = B∗, P2 = B, P3 = B∗ and thus A1 = A3 = B∗, A2 = B and b1 = a2 = a3 = 1,
26
a1 = b2 = b3 = 0. Furthermore, we conclude from the quark content of the three particles thatδη12|1| = δη13|1| = 1, but δη23|1| = 0. With this information we can write down the possible dimers:
d12 = 1×[
1√2
+ 0×(
1− 1√2
)](1× 1 + 0× 0)
[(η12)0 BB∗ + (1− 0) η12BB
∗]
=1√2
(BB∗ + η12BB
∗) (3.14)
d13 = 1×[
1√2
+ 1×(
1− 1√2
)](1× 1 + 0× 0)
[(η13)1 B∗B∗ + (1− 1) η13B
∗B∗]
= η13B∗B∗ (3.15)
d23 = 1× 1×B∗B = B∗B . (3.16)
The next step is now to identify the physical dimers in the system, i.e. those bound or virtualstates which exist in nature (or in the used theory). In the case above one finds on the one handthat there is the Zb(10610) as B∗B bound state and on the other hand the Z ′b(10650) as a B∗B∗
bound state [33]. A BB∗ state is not known so far. Therefore one has to erase d23 from thetheory and only keeps d12 and d13 in a further analysis.
3.1.2 Spin and isospin part of the dimer wave function
Up to now we have ignored the spin and isospin degrees of freedom. Therefore we define projection
operators(O†ij)α,βγ
with combined spin and isospin indices α, β, γ which couple the spin/isospin
of (Ai)β and (Aj)γ to a total dimer spin/isospin of (dij)α. Since the anti-multiplet Ai has the samespin and isospin structure as the multiplet Ai one finds for the dimer wave function includingflavor, spin and isospin:
(dij)α = aiaj (Aj)β
(O†ij)α,βγ
(Ai)γ + bibj(Aj)β
(O†ij)α,βγ
(Ai)γ
+ δηij |1|
[1√2
+ δAiAj
(1− 1√
2
)](biaj + aibj)
×[(ηij)
δAiAj(Aj)β
(O†ij)α,βγ
(Ai)γ +(1− δAiAj
)ηij (Aj)β
(O†ij)α,βγ
(Ai)γ
]
+(1− δηij |1|
)biaj
(Aj)β
(O†ij)α,βγ
(Ai)γ ∀ i < j ∈ 1, 2, 3 . (3.17)
If we again consider the Zb, Z′b example from the previous subsection to clarify the notation, we
would need the two projection operators O†12 and O†13 which are given by(O†12
)α=aA,β=βγ=cγ
= δca−i√
2(τ2τA)βγ ,
(O†13
)α=aA,β=bβγ=cγ
= − 1√2
(Ua)bc−i√
2(τ2τA)βγ , (3.18)
where a, b, c are spin 1 indices, β, γ are isospin 1/2 indices and A is an isospin 1 index. τAare the Pauli matrices and Ua are the generators of the SO(3) rotation group. Hence, Oij is
27
the product of the spin projector and the isospin projector which are completely independentbecause they act in different spaces. Consequently, they commute and the trace in this combinedspin/isospin space simply means ”trace in spin space“ times ”trace in isospin space“. More de-tails on and the derivation of these operators and of all other projectors from 0⊗ 0 up to 1⊗ 1can be found in appendix A.
Following the usual coupling of spins and isospins like 1/2 ⊗ 1/2 = 0 ⊕ 1 we notice that theremight be more than just one dimer with the same particle content. In order to take this fact intoaccount we introduce a second dimer d′ij with identical constituents, but different spin/isospinquantum numbers than dij. Since also the G-parity could change in this case we need in advancea new parameter η′ij and write:
(d′ij)α
= aiaj (Aj)β
(O′†ij)α,βγ
(Ai)γ + bibj(Aj)β
(O′†ij)α,βγ
(Ai)γ
+ δη′ij |1|
[1√2
+ δAiAj
(1− 1√
2
)](biaj + aibj)
×[(η′ij)δAiAj (Aj
)β
(O′†ij)α,βγ
(Ai)γ +(1− δAiAj
)η′ij (Aj)β
(O′†ij)α,βγ
(Ai)γ
]
+(
1− δη′ij |1|)biaj
(Aj)β
(O′†ij)α,βγ
(Ai)γ ∀ i < j ∈ 1, 2, 3 . (3.19)
In general one could continue and define d′′ij, d′′′ij and so on. However, for most two particle systems
two states should be enough since in the end not every possible spin/isospin configuration has abound or virtual state (e.g. in the NN system there are four possible configurations, but onlythe S = 1, I = 0 bound state (the deuteron) and the S = 0, I = 1 virtual state exist in nature).Therefore we restrict ourselves to only one extra dimer d′ij. However, one could straightforwardlyextend this work to a system with three or more states.
3.1.3 Spatial part of the dimer wave function
Finally, we want to make some comments on the spatial part of the dimer wave function. Itdepends on the angular momentum of the constituent particles. Thus, we must specify in whichpartial wave the constituent particles are. Since the energy within a bound state rises with thetotal angular momentum it is more likely that a dimer is formed in the lowest partial wave. Infact, as long as there is no physical property which forbids a L = 0 interaction, it is justified toassume that all dimers are S-wave states. In this work we will ignore the special cases where Lmust be unequal to zero and state that:
all dimers are S-wave states. (3.20)
In this case the spatial structure stays trivial:
dij(x) = aiaj Aj(x) Ai(x) + bibj Aj(x) Ai(x) + ... , (3.21)
and we will not write the x dependence explicitly in our equations. Note, that already for aP -wave dimer there would appear spatial derivatives:
dij(x) = aiaj
[(i~∇Aj(x)
)Ai(x) + Aj(x)
(i~∇Ai(x)
)]+ ... . (3.22)
28
More details on P -wave dimers in EFT can be found in Ref. [129].
3.2 Lagrangian density, vertices and propagators
We are working in a non-relativistic theory. Hence, particles and their corresponding anti-particles are point-like and treated as independent fields. Therefore all particles have the samenon-relativistic kinetic term in the Lagrangian density independently of their spin. Furthermore,there only are contact interactions between the fields due to their point-like character. Asexplained in section 2.1 one can construct an effective Lagrangian density using auxiliary fields.To get a general expression which takes into account all different three particle systems, we needsix auxiliary fields representing the six possible dimers dij and d′ij for i < j ∈ 1, 2, 3. At thispoint one has to choose the order of the Lagrangian density: we will only consider leading order(LO) because already next-to-leading order (NLO) terms depend on the effective range of thedimers which is very poorly known for hadronic molecules so far. In fact, the two nucleon systemis more or less the only system where effective ranges are measured (one exception is the NΛ
system [130,131]). At LO we need six new parameters ∆(′)ij ∈ R and six coupling constants g
(′)ij for
i < j ∈ 1, 2, 3. The ∆ parameters can be related to the scattering length of the correspondingdimer which is at LO related to the – easier measurable – binding momentum via a−1 = γ(see section 1.3). We name the auxiliary fields like the dimers and find Eq. (3.23) where theLagrangian density is given in terms of the isospin multiplet fields A and A. Lines 8 to 12 arethe Hermitian conjugation of lines 3 to 7 and the last line stands for the interaction Lagrangianof d′ij which is achieved by replacing in line 3 to 12 all dij by d′ij, Oij by O′ij, gij by g′ij and ηij byη′ij. From the Lagrangian density one can deduce the vertex factors describing the interactionsbetween the dimers and their constituents as well as the propagators of all included particles.
29
LLO =3∑
i=1
(Ai)†α
(i∂t +
∇2
2mi
)(Ai)α +
3∑
i=1
(Ai)†α
(i∂t +
∇2
2mi
)(Ai)α
+3∑
i, j = 1i < j
(dij)†α ∆ij (dij)α +
3∑
i, j = 1i < j
(d′ij)†α
∆′ij(d′ij)α
−3∑
i, j = 1i < j
(gij
aiaj
(A†i
)β
(Oij)α,βγ(A†j
)γ
(dij)α + bibj
(A†i
)β
(Oij)α,βγ(A†j
)γ
(dij)α
+ δηij |1|
[1√2
+ δAiAj
(1− 1√
2
)](biaj + aibj)
×[(ηij)
δAiAj
(A†i
)β
(Oij)α,βγ(A†j
)γ
(dij)α
+(1− δAiAj
)ηij
(A†i
)β
(Oij)α,βγ(A†j
)γ
(dij)α
]
+(1− δηij |1|
)biaj
(A†i
)β
(Oij)α,βγ(A†j
)γ
(dij)α
+ gij
aiaj
(d†ij
)α
(Aj)γ
(O†ij)α,γβ
(Ai)β + bibj
(d†ij
)α
(Aj)γ
(O†ij)α,γβ
(Ai)β
+ δηij |1|
[1√2
+ δAiAj
(1− 1√
2
)](biaj + aibj)
×[(ηij)
δAiAj
(d†ij
)α
(Aj)γ
(O†ij)α,γβ
(Ai)β
+(1− δAiAj
)ηij
(d†ij
)α
(Aj)γ
(O†ij)α,γβ
(Ai)β
]
+(1− δηij |1|
)biaj
(d†ij
)α
(Aj)γ
(O†ij)α,γβ
(Ai)β
)
−3∑
i, j = 1i < j
(gij → g′ij, dij → d′ij, ηij → η′ij, Oij → O′ij
)(3.23)
30
3.2.1 Vertices
The interaction vertices are obtained by collecting all terms with the same number of fields.Erasing the fields from these terms and multiplying them with a factor i yields the Feynmanrules shown in Fig. 3.1 valid for all indices i, j ∈ 1, 2, 3 with i < j. The two parameters v
(′)ij
and w(′)ij appearing in the vertex factors are defined as
v(′)ij := δ
η(′)ij |1|
[1√2
+ δAiAj
(1− 1√
2
)](biaj + aibj)
(η
(′)ij
)δAiAj+(
1− δη(′)ij |1|
)biaj , (3.24)
and
w(′)ij := δ
η(′)ij |1|
[1√2
+ δAiAj
(1− 1√
2
)](biaj + aibj)
(1− δAiAj
)η
(′)ij . (3.25)
At this point we have to make two remarks. Firstly, note that we write[(Oij)α,γβ
]†=(O†ij)α,βγ
according to projection operators like[(τAτ2)αβ
]†=(
(τAτ2)†)βα
= (τ2τA)βα. The second remark
is on the coupling constants: we use just one symbol gij in front of each interaction term althoughthe coupling strength between e.g. dij and AjAi must not be same as between dij and AjAibecause the dimer is different in both cases. However, in nature dij ∼ PjPi is always uniqueeven if bound states between AjAi and between AjAi exist, since Ai and Ai are not the sameparticle. Thus, it is justified to only use one coupling constant which could have different valuesfor different systems.
3.2.2 Propagators
We can read off from Eq. (3.23) the non-relativistic propagators of the point-like multiplets Aiand Ai:
i (Si)αβ (p0,p) =iδαβ
p0 − |p|2
2mi+ iε
for i = 1, 2, 3 . (3.26)
Hence, all particles propagate forward in time and it is no further work needed. For the dimerfields the procedure is more complicated since they have a coupling to their constituent fields.Firstly, we note that at LO the dimer fields themselves are not dynamic. Their bare propagators,
i(D
(′)0ij
)αβ
(p0,p) = iδαβ
∆(′)ij
, (3.27)
are constant, but they are dressed by loops of their constituent particles. This leads to the Dysonequation in Fig. 3.2(a) and hence to a possible propagation.
31
: −i g(′)ij aiaj
O(′)
ij
α,βγ
Ai
Aj
αβ
γ
d(′)ij : −i g
(′)ij bibj
O(′)
ij
α,βγ
Ai
Aj
αβ
γ
d(′)ij
: −i g(′)ij v
(′)ij
O(′)
ij
α,βγ
Ai
Aj
αβ
γ
d(′)ij : −i g
(′)ij w
(′)ij
O(′)
ij
α,βγ
Ai
Aj
αβ
γ
d(′)ij
β
d(′)ij
α
Ai
Aj γ
: −i g(′)ij aiaj
O(′)†
ij
α,γβ
β
d(′)ij
α
Ai
Aj γ
: −i g(′)ij bibj
O(′)†
ij
α,γβ
β
d(′)ij
α
Ai
Aj γ
: −i g(′)ij v
(′)ij
O(′)†
ij
α,γβ
β
d(′)ij
α
Ai
Aj γ
: −i g(′)ij w
(′)ij
O(′)†
ij
α,γβ
Figure 3.1: Vertex factors for all possible interaction terms depending on the parameters a andb valid ∀ i, j ∈ 1, 2, 3 with i < j. Time and momentum flow from left to right and furthermore
v(′)ij and w
(′)ij are given in Eq. (3.24) and Eq. (3.25).
=α β α β
+ ...+α β
Ai
Aj
+α β
Ai
Aj
+α β
Ai
Aj
+α β
Ai
Aj
(a)
= + + +iΣ
(′)ij
αβ
(p0,p) α βγ
σα β
γ
σα β
γ
σα β
γ
σ
Ai
Aj
Ai
Aj
Ai
Aj
Ai
Aj
(b)
Figure 3.2: Diagrammatic representation of the Dyson equation. The full dimer propagator isdepicted as double line and the bare ones as thick solid lines (a). Diagrammatic representationof the dimer self-energy (b).
32
Introducing the self-energy(
Σ(′)ij
)αβ
(p0,p) the full dimer propagator i(D
(′)ij
)αβ
(p0,p) is given
by
i(D
(′)ij
)αβ
= i(D
(′)0ij
)αβ
+ i(D
(′)0ij
)αγ
i(
Σ(′)ij
)γρi(D
(′)0ij
)ρβ
+ i(D
(′)0ij
)αγ
i(
Σ(′)ij
)γρi(D
(′)0ij
)ρµi(
Σ(′)ij
)µνi(D
(′)0ij
)νβ
+ ...
= i(D
(′)0ij
)αγ
[(1)γβ + i
(Σ
(′)ij
)γρi(D
(′)0ij
)ρβ
+ i(
Σ(′)ij
)γρi(D
(′)0ij
)ρµi(
Σ(′)ij
)µνi(D
(′)0ij
)νβ
+ ...
]
= i(D
(′)0ij
)αγ
∞∑
n=0
[−(
Σ(′)ij
)γρ
(D
(′)0ij
)ρβ
]n. (3.28)
Self-energy
In the next step one needs to determine the self-energy. Therefore we consider all Feynmandiagrams which – depending on the parameters ai and bi – contribute to it (Fig. 3.2(b)) andconclude:
i(
Σ(′)ij
)αβ
(p0,p) =(g
(′)ij
)2
Sij
(O(′)ij
)α,γσ
(O(′)†ij
)β,σγ
[aiajaiaj + v
(′)ij v
(′)ij + w
(′)ij w
(′)ij + bibjbibj
]
×∫
d4q
(2π)4
1
p0 + q0 − 12mi
(p2
+ q)2
+ iε
1
−q0 − 12mj
(p2− q
)2+ iε
, (3.29)
with loop momentum (q0,q) and Sij being the symmetry factor of the diagrams shown inFig. 3.2(b). Depending on whether Pi is identical to Pj or not the latter is either 1 or 2 (seeappendix B):
Sij =
2 , if Pi = Pj
1 , if Pi 6= Pj. (3.30)
Before we continue it is useful to have a closer look on the term in square brackets: since Picannot be equal to Ai and Ai at once, we know that there only are two possible parameter setsai = 1, bi = 0 or ai = 0, bi = 1. Combining this with the set of particle Pj and taking intoaccount a possible non-vanishing G-parity quantum number, five physical combinations for theterm in square brackets remain:
• ai = aj = 1 ∧ bi = bj = 0 ∧ η(′)ij = 0
⇒[(aiaj) +
(v
(′)ij
)2
+(w
(′)ij
)2
+ (bibj)2
]= aiaj = 1
• ai = aj = 0 ∧ bi = bj = 1 ∧ η(′)ij = 0
⇒[(aiaj) +
(v
(′)ij
)2
+(w
(′)ij
)2
+ (bibj)2
]= bibj = 1
33
• bi = aj = 1 ∧ ai = bj = 0 ∧ η(′)ij = 0
⇒[(aiaj) +
(v
(′)ij
)2
+(w
(′)ij
)2
+ (bibj)2
]= biaj = 1
• bi = aj = 1 ∧ ai = bj = 0 ∧ η(′)ij = ±1
⇒[(aiaj) +
(v
(′)ij
)2
+(w
(′)ij
)2
+ (bibj)2
]= biaj = 1
• bi = aj = 0 ∧ ai = bj = 1 ∧ η(′)ij = ±1
⇒[(aiaj) +
(v
(′)ij
)2
+(w
(′)ij
)2
+ (bibj)2
]= δ
η(′)ij |1|
aibj = 1
Hence, one finds
[(aiaj) +
(v
(′)ij
)2
+(w
(′)ij
)2
+ (bibj)2
]= 1 ∀ ai, bi . (3.31)
Consequently, the self-energy and therefore the full propagator of the dimer d(′)ij are independent
of the species of its constituents. Thus, the self-energy simplifies to
i(
Σ(′)ij
)αβ
(p0,p) =(g
(′)ij
)2
Sij
(O(′)ij
)α,γσ
(O(′)†ij
)β,σγ
×∫
d4q
(2π)4
1
p0 + q0 − 12mi
(p2
+ q)2
+ iε
−1
q0 + 12mj
(p2− q
)2 − iε. (3.32)
Using the residue theorem one can perform the integration over q0 which yields after some algebra
i(
Σ(′)ij
)αβ
(p0,p) =(g
(′)ij
)2
Sij
(O(′)ij
)α,γσ
(O(′)†ij
)β,σγ
×∫
d3q
(2π)3
2µiji
q2 − 2µij p0 + p2
4+√
1− 4µijmi+mj
p · q− iε, (3.33)
where we have written the modulus of a 3-vector as |x| := x and introduced the reduced mass
µij :=mi mj
mi +mj
, (3.34)
which is obviously equal for both unprimed and primed dimers. The remaining integral can becalculated in dimensional regularization using a scale µ. As one can found in many textbookson QFT (e.g. Ref. [1]) it holds
∫ddq
1
(q2 + 2 q · k − b2)α= (−1)
d2 iπ
d2
Γ(α− d
2
)
Γ(α)
[−k2 − b2
] d2−α
, (3.35)
34
which was also applied to the self-energy integral in section 2.1. Thus, in D = 4 dimensions wehave with d = D − 1:(
Σ(′)ij
)αβ
=(g
(′)ij
)2
Sij
(O(′)ij
)α,γσ
(O(′)†ij
)β,σγ
× µD−4
∫dD−1q
(2π)D−1
2µij
q2 − 2µij p0 + p2
4+√
1− 4µijmi+mj
p · q− iε
= 2(g
(′)ij
)2
µij Sij
(O(′)ij
)α,γσ
(O(′)†ij
)β,σγ
µD−4
(2π)D−1
[(−1)
D−12 i π
D−12
Γ(α− D−1
2
)
Γ(1)
×−(
1− 4µijmi +mj
)p2
4−(
2µij p0 −p2
4+ iε
)D−12−1]
D→4−→ − 1
2π
(g
(′)ij
)2
µij Sij
(O(′)ij
)α,γσ
(O(′)†ij
)β,σγ
√−2µij
(p0 −
p2
2(mi +mj)
)− iε .
(3.36)
In the last step we took the limit D → 4. This is possible because there is no pole in D = 4dimensions. However, for D = 3 it is and therefore the S-wave scattering length has an unnaturalscaling behavior as it is discussed for nucleon–nucleon interactions in Ref. [70]. As the sameauthors pointed out in Ref. [72], a scattering length of natural size scales as anat ∼ 1/µ sincethe scale µ is set by the pion mass where EFT(/π) breaks down. However, the unnatural largescattering length aNN in the NN system yields aNN 1/µ. This fact will lead to problems inthe power counting scheme of the EFT. Hence, they introduced in Refs. [71, 72] instead of theminimal subtraction scheme which only removes poles in four dimensions, a new scheme calledpower divergence subtraction (PDS) scheme which subtracts the poles in both three and four
dimensions by a counter term δΣ(′)ij so that a correct power counting is restored. Although the
authors only assumed NN interactions the same argument is true for any other system withunnatural large scattering length. Hence, one should use the PDS scheme also for hadronicmolecules whose scattering length is expected to be rather large. With the PDS scale µPDS(which we use instead of µ) the corresponding PDS counter term for our problem is given by
(δΣ
(′)ij
)αβ
=1
2π
(g
(′)ij
)2
µij Sij
(O(′)ij
)α,γσ
(O(′)†ij
)β,σγ
µPDS
D − 3, (3.37)
and the self-energy is changed to(
Σ(′)ij
)αβ
PDS−→(
Σ(′)ij
)αβ
(p0,p) +(δΣ
(′)ij
)αβ
= 2(g
(′)ij
)2
µij Sij
(O(′)ij
)α,γσ
(O(′)†ij
)β,σγ
µD−4PDS
(2π)D−1
[(−1)
D−12 i π
D−12
Γ(α− D−1
2
)
Γ(1)
×−(
1− 4µijmi +mj
)p2
4−(
2µij p0 −p2
4+ iε
)D−12−1]
+1
2π
(g
(′)ij
)2
µij Sij
(O(′)ij
)α,γσ
(O(′)†ij
)β,σγ
µPDS
D − 3, (3.38)
35
which yields after taking the limit D → 4
(Σ
(′)ij
)αβ
D→4−→ − 1
2π
(g
(′)ij
)2
µij Sij
(O(′)ij
)α,γσ
(O(′)†ij
)β,σγ
√−2µij
(p0 −
p2
2(mi +mj)
)− iε
+1
2π
(g
(′)ij
)2
µij Sij
(O(′)ij
)α,γσ
(O(′)†ij
)β,σγ
µPDS
=1
2π
(g
(′)ij
)2
µij Sij
(O(′)ij
)α,γσ
(O(′)†ij
)β,σγ
×[−√−2µij
(p0 −
p2
2(mi +mj)
)− iε+ µPDS
]. (3.39)
Before we proceed we shorten this result via the definition
(Σ
(′)ij
)(p0,p) :=
1
2π
(g
(′)ij
)2
µij Sij
[−√−2µij
(p0 −
p2
2(mi +mj)
)− iε+ µPDS
], (3.40)
and write:(
Σ(′)ij
)αβ
(p0,p) =(O(′)ij
)α,γσ
(O(′)†ij
)β,σγ
(Σ
(′)ij
)(p0,p) . (3.41)
Now consider the (iso)spin dependent part(O(′)ij
)α,γσ
(O(′)†ij
)β,σγ
. It is not necessary to know
the projection operators in detail: as shown in appendix A all projectors are orthonormal (or atleast orthogonal if one has shifted the normalization factor into the coupling constant within theLagrangian Eq. (3.23)). Consequently, it holds
(O(′)ij
)α,γσ
(O(′)†ij
)β,σγ
=
((O(′)ij
)α
(O(′)†ij
)β
)
γγ
= Tr
((O(′)ij
)α
(O(′)†ij
)β
)= c
(′)ij δαβ , (3.42)
with c(′)ij ∈ R. If – as in our case (see appendix A) – all projectors are correctly normalized one
finds that
if O(′)ij is normalized: c
(′)ij = 1 ∀ i < j ∈ 1, 2, 3 . (3.43)
Otherwise, c(′)ij depends on the in this case changed prefactors in the projection operators and
must be calculated separately. Note, that such a different normalization would be canceled atanother point in the calculation and does not change any observable. Considering again ourexample of Zb(10610) and Z ′b(10650) we find (cf. Eq. (3.18)):
(O12)α=aA,γ=cγσ=σ
(O†12
)β=bB,σ=σγ=cγ
= δaci√2
(τAτ2)γσ δcb−i√
2(τ2τB)σγ
=1
2δab Tr (τAτ2τ2τB)
=1
2δab Tr (τAτB)
= δab δAB , (3.44)
36
(O13)α=aA,γ=cγσ=sσ
(O†13
)β=bB,σ=sσγ=cγ
=−i2
(Ua)cs (τAτ2)γσi
2(Ub)sc (τ2τB)σγ
=1
4Tr (UaUb) Tr (τAτ2τ2τB)
= δab δAB , (3.45)
as expected since O12 and O13 are indeed normalized. However, we will keep c(′)ij in our work to
be as general as possible. Inserting Eq. (3.42) into Eq. (3.41) yields(
Σ(′)ij
)αβ
(p0,p) = δαβ c(′)ij Σ
(′)ij (p0,p) . (3.46)
Together with Eq. (3.27) we can use this result to simplify Eq. (3.28) to
i(D
(′)ij
)αβ
(p0,p) = iδαγ
∆(′)ij
∞∑
n=0
[−δγρ c(′)
ij Σ(′)ij (p0,p)
δρβ
∆(′)ij
]n
= iδαγ
∆(′)ij
∞∑
n=0
(δγβ)n[−c(′)
ij Σ(′)ij (p0,p)
1
∆(′)ij
]n
= iδαγ
∆(′)ij
∞∑
n=0
δγβ
[−c(′)
ij Σ(′)ij (p0,p)
1
∆(′)ij
]n
= δαβi
∆(′)ij
∞∑
n=0
[−c(′)
ij
(Σ
(′)ij
)(p0,p)
1
∆(′)ij
]n
= δαβi
∆(′)ij
1
1−(−c(′)
ij Σ(′)ij (p0,p) 1
∆(′)ij
)
=iδαβ
∆(′)ij + c
(′)ij Σij(p0,p)
, (3.47)
where we have used the geometric series in the second to last step. Using the definition of Σij inEq. (3.40) we finally find that the full dimer propagator is given by
i(D
(′)ij
)αβ
(p0,p) = − 2π i(g
(′)ij
)2
µij Sij c(′)ij
× δαβ
−(
2π ∆(′)ij(
g(′)ij
)2µij Sij c
(′)ij
+ µPDS
)+
√−2µij
(p0 − p2
2(mi+mj)
)− iε
.
(3.48)
This result can be used to determine the elastic two-body scattering amplitude T elij of the particles
Pi and Pj. The advantage is then that one can compare T elij with the LO effective range expansion
37
amplitude TEREij in order to write the full dimer in terms of an observable, namely the binding
momentum γ(′)ij instead of the parameter ∆
(′)ij .
Elastic scattering amplitude
In Fig. 3.3 we consider all possible diagrams which could – depending on the parameters a and
b – contribute to the elastic on-shell scattering amplitude T(′)elij (E = k2/(2µij), 0) of the two
particles Pi and Pj. Written as an equation we get
i(T
(′)elij
)µναβ
(E =
k2
2µij, 0
)= −
(g
(′)ij
)2
Sel
[aiajaiaj + v
(′)ij v
(′)ij + w
(′)ij w
(′)ij + bibjbibj
]
×(O(′)ij
)σ,µν
i(D
(′)ij
)σγ
(E =
k2
2µij, 0
)(O(′)†ij
)γ,βα
=2π i
µij c(′)ij
SelSij
(O(′)ij
)γ,µν
(O(′)†ij
)γ,βα
× 1
−(
2π ∆(′)ij(
g(′)ij
)2µij Sij c
(′)ij
+ µPDS
)+√−k2 − iε
, (3.49)
with symmetry factor Sel (see appendix B),
Sel =
4 , if Pi = Pj
1 , if Pi 6= Pj, (3.50)
and in the second step we have used that the term in square brackets is equal to 1 (cf. Eq. (3.31)).Due to the −iε in the square root one has to choose its negative branch cut and we end up witha relation only depending on k:
(T
(′)elij
)µναβ
(k) = − 2π
µij c(′)ij
SelSij
(O(′)ij
)γ,µν
(O(′)†ij
)γ,βα
1(2π ∆
(′)ij(
g(′)ij
)2µij Sij c
(′)ij
+ µPDS
)+ ik
. (3.51)
However, it still has some free (iso)spin indices so we apply projection operators to it. In anelastic scattering process we know that initial and final state (iso)spin must be equal to the
iT
(′)elij
µν
αβ
E = k2
2µij, 0
= +
d(′)ij
Ai
Aj
Ai
Aj
γ σ
α
β
µ
ν
+d(′)ij
Ai
Aj
Ai
Aj
γ σ
α
β
µ
νd(′)ij
Ai
Aj
Ai
Aj
γ σ
α
β
µ
ν
Ai
+d(′)ij
Ai
Aj Aj
γ σ
α
β
µ
ν
Figure 3.3: Diagrams contributing to the elastic on-shell scattering amplitude of the two par-ticles Pi and Pj which correspond to one of the four possible combinations of the multiplets A,characterized by the parameters a and b.
38
(iso)spin of the intermediate dimer. Therefore we conclude that the needed projectors are thesame as those for the vertices within the diagrams in Fig. 3.3. From Eq. (3.42) we know that
O(′)ij is not necessarily normalized and thus we have to do this now via a factor 1/
√c
(′)ij . Note,
that for already normalized projectors c(′)ij = 1 and thus nothing is changed by the extra factor.
After projection onto the dimer (iso)spin one still has to average over initial and sum over final(iso)spins in order to obtain an index-free amplitude:
T(′)elij =
1
dof
∑
η,ρ
(T
(′)elij
)ρη
=1
dof
∑
η,ρ
1√c
(′)ij
(O(′)†ij
)ρ,νµ
(T
(′)elij
)µναβ
1√c
(′)ij
(O(′)ij
)η,αβ
. (3.52)
Here dof stands for ”spin degrees of freedom × isospin degrees of freedom“, i.e. dof = (2J + 1)×(2I + 1). Inserting what we calculated in Eq. (3.51) leads to
T(′)elij =
1
dof
∑
η,ρ
−2π
µij
(c
(′)ij
)2
SelSij
Tr
((O(′)†ij
)ρ
(O(′)ij
)γ
)Tr
((O(′)†ij
)γ
(O(′)ij
)η
)
× 1(2π ∆
(′)ij(
g(′)ij
)2µij Sij c
(′)ij
+ µPDS
)+ ik
. (3.53)
Following Eq. (3.42) both traces yield a factor c(′)ij and a Kronecker-delta for the (iso)spin indices:
T(′)elij = − 2π
µij
SelSij
1(2π ∆
(′)ij(
g(′)ij
)2µij Sij c
(′)ij
+ µPDS
)+ ik
1
dof
∑
η,ρ
δρη . (3.54)
The sum over the (iso)spin indices ρ and η reduces to∑
η,ρ
δρη = δηη = Tr(1dim. spin space
)Tr(1dim. isospin space
)
= dim. spin space× dim. isospin space
= spin degrees of freedom× isospin degrees of freedom
= dof . (3.55)
Thus, we end up with
T(′)elij = − 2π
µij
SelSij
1(2π ∆
(′)ij(
g(′)ij
)2µij Sij c
(′)ij
+ µPDS
)+ ik
, (3.56)
which can be compared to the ERE amplitude (cf. section 1.4),
T(′)EREij = −
(2 + 2 δPiPj
)π
µij
11
a(′)ij
+ ik, (3.57)
39
where the additional term 2 δPiPj ensures the right normalization in case of identical particles.Using the definitions of Sel (Eq. (3.50)) and Sij (Eq. (3.30)) we observe that also
SelSij
=
42
= 2 , if Pi = Pj11
= 1 , if Pi 6= Pj(3.58)
reproduces the right prefactor in both cases. Therefore we can identify the scattering length a(′)ij
which is at LO equivalent to the binding momentum γ(′)ij as
1
a(′)ij
≡ γ(′)ij =
2π ∆(′)ij(
g(′)ij
)2
µij Sij c(′)ij
+ µPDS . (3.59)
Note, that the explicit dependence on the PDS scale µPDS may be shifted into the couplingconstant, but for our purpose this does not cause any problems. Hence, we leave the equationabove unchanged. Furthermore, we can plug it into Eq. (3.48) to find an expression for the LO
full dimer propagator which only depends on observables except for the prefactor(g
(′)ij
)−2
which
is canceled anyway in all following calculations. We find:
i(D
(′)ij
)αβ
(p0,p) = − 2π i(g
(′)ij
)2
µij Sij c(′)ij
δαβ
−γ(′)ij +
√−2µij
(p0 − p2
2(mi+mj)
)− iε
:= δαβ iD(′)ij (p0,p) , (3.60)
where we have defined an expression without (iso)spin indices in the last step.
3.2.3 Wave function renormalization constant
The only missing relation before one can derive a general d(′)ij –Pk scattering amplitude is the wave
function renormalization constant Z(′)ij . Firstly, we note that the fields Ai and Ai have a wave
function renormalization constant equal to one since they are considered as point-like particles.Only the dimer fields must be renormalized. From basic QFT we know that Z
(′)ij is the residue
of the corresponding (full) propagator [1]:(Z
(′)ij
)−1
= i∂
∂p0
[(iD
(′)ij (p0,p)
)−1]∣∣∣∣p0=−(γ(′)ij )
2
2µij, p=0
, (3.61)
where p0 is set to the binding energy B(′)ij of the bound or virtual state characterized by the
dimer. As explained in section 1.4.1 we use the definition
γ(′)ij ≡ sgn
(B
(′)ij
)√2µij
∣∣∣B(′)ij
∣∣∣ , (3.62)
for the binding momentum γ(′)ij . With the result of Eq. (3.60) we find
Z(′)ij =
2π γ(′)ij(
g(′)ij
)2
µ2ij Sij c
(′)ij
. (3.63)
40
3.3 General three-body scattering amplitude
In principle one has to analyze d12–P3, d13–P2, d23–P1, d′12–P3, d′13–P2 and d′23–P1 scattering inorder to account for all possible three particle states. Alternatively, but much simpler one canonly calculate d12–P3 scattering and instead change – if necessary – the particle allocation in thefollowing way:
Particle allocation
1. Choose the three particles of the system (→ e.g. B, B∗, B∗).
2. Choose the dimer–particle scattering process you are interested in within this system (→e.g.
(BB∗
)–B∗).
3. Allocate the multiplets according to that choice, but always in a way that P3 is scatteredoff the dimer d12 (→ e.g. A1 = B, A2 = B∗, A3 = B∗).
4. Set ai, bi in Pi (i ∈ 1, 2, 3) to either 0 or 1 so that the desired three particle system isreproduced (→ e.g. a1 = b2 = a3 = 1, b1 = a2 = b3 = 0 ⇒ P1 = B, P2 = B∗, P3 = B∗).
d(′)ij
Pk
d(′)ij
Pk
T(′)ij
iT(′)ij =
Figure 3.4: Fixing the notation for the dimer–particle scattering amplitudes T(′)ij with i, j, k ∈
1, 2, 3 and i < j, k 6= i, j.
The notation for the mentioned amplitudes is shown in Fig. 3.4. Although with this schemewe only need to determine d12–P3 scattering it is in general possible that all other scatteringamplitudes contribute to this scattering. Hence, in most cases one has to solve a coupled integralequation system of all amplitudes. The Feynman diagrams for all six coupled amplitudes areshown in Fig. C.2 and explained in appendix C. The straightforward, but lengthy task is nowto use the Feynman rules derived in the previous section 3.2 and to determine the coupledintegral equation system. To do so we once more have to fix our notation for (iso)spin indicesand momenta. We will work in the center-of-mass system with incoming 4-momenta k, −kand outgoing 4-momenta p, −p. Omitting possible ”primes“ for the moment the center-of-massenergy is
E =k2
2(mi +mj)+
k2
2mk
− γij2µij
. (3.64)
Furthermore, we assign the combined spin and isospin indices as follows:
41
• initial state dimer: α,
• initial state single particle: β,
• final state dimer: γ,
• final state single particle: σ,
• intermediate dimer: µ,
• intermediate single particle: ν,
• intermediate exchanged particle: ρ.
Since there only are two different topologies for the Feynman diagrams the notation above leadsto the assignment shown in Fig. 3.5 which is valid for every diagram we need to determine. Inappendix B we have derived the symmetry factor for diagrams like those in Figure 3.5.
(k2
2(mi +mj)− γij
2µij,k
)
(k2
2mk,−k
)
α
β
σ
γ
E − k2
2mk− p2
2mi
k+ q
(p2
2mi,−p
)
(p2
2(mj +mk)− γjk
2µjk,p
)
α
β
σ
γ
(k2
2(mi +mj)− γij
2µij,k
)
(k2
2mk,−k
)
(p2
2mk,−p
)
(p2
2(mi +mj)− γij
2µij,p
)
E − p2
2mk− q2
2mi
q+ p
(q0,q)
(−q0,−q)
T
ρ
ρµ
ν
Figure 3.5: Definition of (iso)spin indices and momenta in the center-of-mass system for bothdiagram topologies appearing in the scattering amplitudes.
We found
Sijk = ζijk(1 + δPiPj + δPjPk + δPiPj δPjPk
), (3.65)
with ζijk defined in Eq. (B.14).With all these ingredients one finally can write down the coupled integral equation system ofthe six amplitudes contributing to the general d12–P3 scattering amplitude corresponding to thediagrams shown in Figure C.2. For details on the equations see appendix F. In Eq. (F.1) wehave combined all diagrams with the same momentum structure. Furthermore, we added termswhich only differ by a ”primed“ or ”unprimed“ final state dimer. In order to have more compact
42
equations we have introduced two short-hand notations f(h)
(ij)(′)(kn)[′] and f(h)
(ij)(′)(kn)[′] for the sum
over the ai, bi dependent vertex factors:
f(h)
(ij(′))(kn[′]):=
aiajakan + v
(′)ij akan + ahw
(′)ij v
[′]kn + bibjv
[′]kn + aiajw
[′]kn + bhv
(′)ij w
[′]kn + w
(′)ij bkbn
aiajakan + v(′)ij akan + ahw
(′)ij w
[′]kn + bibjw
[′]kn + aiajv
[′]kn + bhv
(′)ij v
[′]kn + w
(′)ij bkbn
+ bibjbkbn , for i < k ∧ j = 2, k = 3 ∨ j = 3, k = 2+ bibjbkbn , else
, (3.66)
f(h)
(ij(′))(kn[′]):=
aiajakan + v
(′)ij bkbn + bhw
(′)ij w
[′]kn + aiajw
[′]kn + bibjv
[′]kn + ahv
(′)ij v
[′]kn + w
(′)ij akan
aiajakan + v(′)ij bkbn + bhw
(′)ij v
[′]kn + aiajv
[′]kn + bibjw
[′]kn + ahv
(′)ij w
[′]kn + w
(′)ij akan
+ bibjbkbn , fori < k ∧ j = 2, k = 3 ∨ j = 3, k = 2
∨ i = k ∧ j = n
+ bibjbkbn , else,
(3.67)
where it was necessary to replace the vertex factor w(′)ij by w
(′)ij := w
(′)ij + δAiAj v
(′)ij because a
diagram like that in Fig. 3.6 yields in general a vertex factor proportional to w13. However, if theconstituents A1 and A3 are identical (although P1 = A1 6= A3 = P3) one obtains according to thevertex rules in Fig. 3.1 a factor v13 instead. In Eq. (F.1) the q0 integration is already performed
d13
A1
A2
T12
d12
A3
d12
P3
Figure 3.6: Exemplary diagram to clarify the change of the vertex factor w(′)ij → w
(′)ij := w
(′)ij +
δAiAj v(′)ij which is necessary for not missing out contributions to the molecule–particle scattering
amplitudes.
using the residue theorem: after applying the Feynman rules the terms have the form∫
d4q
(2π)4t(E,k,q)
1
−q0 − q2
2mi+ iε
D(E + q0,q)
E + q0 − p2
2mj− (p+q)2
2mk+ iε
=:
∫d3q
(2π)3
∫dq0
2πϕ(q0) . (3.68)
They are simplified by noting that ϕ(q0) has a (` = 1)-fold pole at q0 = −q2/2mi + iε. Thereforewe used the residue theorem,∫dq0
2πϕ(q0) = iξResϕ
(− q2
2mi
+ iε
), where Resg(z)(c) =
1
(`− 1)!
∂`−1
∂z`−1
[(z − c)`g(z)
]∣∣∣∣z=c
,
with positive winding number ξ = +1 to obtain that Eq. (3.68) is reduced to
−i∫
d3q
(2π)3t(E,k,q)D
(E − q2
2mi
,q
)1
E − q2
2mi− p2
2mj− (p+q)2
2mk+ iε
, (3.69)
43
which is the form appearing in Eq. (F.1). However, neither (iso)spin projection nor wave function
renormalization is applied to the amplitudes(t(′)ij
)γσαβ
. As explained in appendix C there are
factors (1 − δPiPj/2) in front of each diagram and thus in front of each amplitude in Eq. (F.1)which are chosen according to the rules concerning identical particles explained in appendix Cand reviewed in the box on page 51.
3.3.1 Wave function renormalization
We proceed by applying wave function renormalization to the amplitudes in Eq. (F.1) by multi-plying each amplitude with the square root of the wave function renormalization constant of theincoming dimer and with the square root of the wave function renormalization constant of theoutgoing dimer:
(T12
)γσαβ(
T13
)γσαβ(
T23
)γσαβ(
T ′12
)γσαβ(
T ′13
)γσαβ(
T ′23
)γσαβ
:=√Z12
(t12)γσαβ√Z12
(t13)γσαβ√Z13
(t23)γσαβ√Z23
(t′12)γσαβ√Z ′12
(t′13)γσαβ√Z ′13
(t′23)γσαβ√Z ′23
, (3.70)
with Z(′)ij given in Eq. (3.63). The renormalized (i.e. physical) amplitudes T
(′)ij are then propor-
tional to√γ12/γ
(′)ij . To get rid of these factors one can redefine all amplitudes via
T(′)ij :=
√γ12
γ(′)ij
T(′)ij . (3.71)
Note, that the d12–P3 scattering amplitude T12 = T12 in which we are interested is not changedby this definition. Thus, we can divide both sides of Eq. (F.1) by i and plug in the full dimerpropagator Eq. (3.60) to find Eq. (F.2) given in appendix F.
3.3.2 Projection on partial wave amplitude
One would expect that the scattering of a particle P3 off the dimer d12 preferentially occurs in thepartial wave with L = 0 because there is no general reason which forbids a S-wave interactionand one expects the energy of the scattering state to be minimal. Nevertheless, we keep theequations general and project out the L-th partial wave amplitude T
(′)(L)ij in Eq. (F.2). For
details on this projection see appendix D, especially for the properties of the Legendre functionof the second kind QL which we use in the following analysis. As a remainder we once more give
44
its definition:
QL(β − iε) := (−1)L1
2
∫ 1
−1
dxPL(x)
(β − iε) + xwith β ∈ R and |β| 6= 1 . (3.72)
Using Eq. (D.18) one can rewrite all six amplitudes T(′)ij (E,k,p) in terms of the partial wave
projected amplitudes T(′)(L)ij (E, k, p). The corresponding results can be found in appendix F in
Eq. (F.3) where we have introduced a short-hand notation for the repeatedly appearing Legendrefunction:
QijkL (q, p;E) := QL
(mi
qp
(q2
2µij+
p2
2µjk− E
)− iε
), (3.73)
where one has to keep in mind that µij = µji holds by definition.
3.3.3 Spin and isospin projection onto specific scattering channel
Now we can concentrate on the spin and isospin structure: we start by applying projectionoperators O
T(′)ij
to the amplitudes on the left-hand-side of Eq. (F.3) which couple the (iso)spins
of the dimer d(′)ij and the third particle Pk to an incoming (iso)spin state η and an outgoing one
λ:
(T
(′)ij
)λη
= (OT12)η,αβ(T
(′)ij
)γσαβ
(O†T
(′)ij
)
λ,σγ
. (3.74)
Note here, that in general the projection operators OTij and Oij are not the same because theformer couples dimer and particle (iso)spin while the latter couples two particle (iso)spins to thedimer (iso)spin. To get the full scattering amplitude one still has to average over initial (iso)spinand to sum over the final one:
T(′)ij =
1
dof(η)
∑
η,λ
(T
(′)ij
)λη
=1
dof(η)
∑
η,λ
(OT12)η,αβ(T
(′)ij
)γσαβ
(O†T
(′)ij
)
λ,σγ
. (3.75)
Similar to what we have done in section 3.2.2 ”dof(η)“ represents the spin degrees of freedom timesthe isospin degrees of freedom in the initial state. If we only consider the (iso)spin dependentterms in Eq. (F.3) the application of the projection operators yields according to Eq. (3.75) the
45
structure below:
T12
T13
T23
T ′12
T ′13
T ′23
=1
dof(η)
∑
η,λ
(OT12)η,αβ
(T12)γσαβ(T13)γσαβ(T23)γσαβ(T ′12)γσαβ(T ′13)γσαβ(T ′23)γσαβ
(O†T12)λ,σγ(O†T13)λ,σγ(O†T23)λ,σγ(O†T ′12)λ,σγ(O†T ′13)λ,σγ(O†T ′23)λ,σγ
∼
x1
y1
z1
x′1y′1z′1
+
x1
y1
z1
x′1y′1z′1
+
x2
y2
z2
x′2y′2z′2
T12 +
x2
y2
z2
x′2y′2z′2
T12 +
x3
y3
z3
x′3y′3z′3
T13 +
x3
y3
z3
x′3y′3z′3
T13 +
x4
y4
z4
x′4y′4z′4
T23 +
x4
y4
z4
x′4y′4z′4
T23
+
x5
y5
z5
x′5y′5z′5
T ′12 +
x5
y5
z5
x′5y′5z′5
T ′12 +
x6
y6
z6
x′6y′6z′6
T ′13 +
x6
y6
z6
x′6y′6z′6
T ′13 +
x7
y7
z7
x′7y′7z′7
T ′23 +
x7
y7
z7
x′7y′7z′7
T ′23 , (3.76)
where we have introduced new parameters x(′)i , y
(′)i , z
(′)i , x
(′)i , y
(′)i and z
(′)i for i ∈ 1, 2, 3, 4, 5, 6, 7.
Their definition (Eq. (A.40) - Eq. (A.81)) and the derivation of the equation above can be foundin appendix A.2. In the last step we want to combine the pairs of terms proportional to the sameamplitude T
(′)ij . We know if δPiPj = 1 then
• mi = mj ⇒ µik = µjk,
• v(′)ij = w
(′)ij = 0 because Pi = Pj directly induces that there are no mixed terms of ai and bj,
i.e. only ai = aj = 1 or bi = bj = 1 are possible combinations,
• δPiPjf (m)(ij)(kn) = δPiPj f
(m)(ij)(kn) since for i < k ∧ j = 2, k = 3 ∨ j = 3, k = 2 (and similarly
for the ”else“ case in the definition of f and f) it holds:
δPiPj
(aiajakan + vijakan + amwijvkn + bibjvkn + aiajwkn + bmvijwkn + wijbkbn + bibjbkbn
)
= δPiPj
(aiajakan + bibjvkn + aiajwkn + bibjbkbn
)
= δPiPj
(aiajakan + amvijvkn + wijakan + bibjvkn + aiajwkn + vijbkbn + bmwijwkn + bibjbkbn
).
Therefore we can factor out not only the amplitude itself, but also the a, b parameter dependentterm and write the amplitudes of the coupled integral equation system for d12–P3 scattering asshown in Eqs. (F.4 - F.9).
46
3.4 Asymptotic behavior
The momentum integrals in the previously mentioned Eqs. (F.4 - F.9) are divergent. Conse-quently, we introduce a momentum cutoff ΛC to regularize them. In principle, one could proceedand solve the coupled integral equation system numerically for different three particle systemsin order to find a possible Efimov effect. However, this would mean that we loose all generalityin our equations. More favorably would be a method which keeps this generality. Indeed, onecan consider the approach of asymptotic large off-shell momenta ΛC q, p γ
(′)ij ∼ k ∼ E for
i < j ∈ 1, 2, 3 which has this property. In Ref. [80] and in the references therein it is explainedhow one can use asymptotic Faddeev wave functions to derive the conditions under which theEfimov effect can occur in a system. However, this method does not provide an instruction to– more or less – directly read off from the properties of the three particles whether the Efimoveffect occurs if the particles have more general spin and isospin quantum numbers and differentscattering channels. The equations themselves are not changed that much considering these ad-ditional degrees of freedom, but to determine all the parameters therein becomes more involvedfor higher spins and isospins. In contrast one can – still using the asymptotic momentum ap-proach – consider the work in Ref. [166] and follow its applications in Refs. [92, 167]. Here, theintegral equation system is decoupled in order to reduce the problem to the determination ofeigenvalues of a matrix whose entries are spin and isospin dependent. To do so we will see thatone needs to make some assumptions concerning the ratio of masses within the three particlesystem. However, these assumptions will match with the given systems if we consider particlesscattering off hadronic molecules. Therefore we will use the second method.In the mentioned limit of large asymptotic off-shell momenta both, the denominator of the inte-grand and the argument of the Legendre function QL considerably simplify. Taking into accountthat one has to take the limit ε→ 0 in the end, it holds
limε→0
− γ(′)ij +
√−2µij
(E − q2
2mk
− q2
2(mi +mj)
)− iε
qE−→ limε→0
− γ(′)ij +
√q2µij
(1
mk
+1
mi +mj
)− iε
= −γ(′)ij + q
õij
(1
mk
+1
mi +mj
)
qγij−→√
µijµ(ij)k
q , (3.77)
where we have introduced the reduced mass of the dimer d(′)ij and particle Pk system,
µ(ij)k =(mi +mj)mk
mi +mj +mk
. (3.78)
47
Moreover, – omitting the particle indices for the moment – the argument of the Legendre functionreduces to
limε→0
m
qpQL
(m
qp
(q2
2µ+p2
2µ− E
)− iε
)p,qE−→ lim
ε→0
m
qpQL
(m
qp
(q2
µ+p2
2µ
)− iε
)
=m
qpQL
(m
qp
(q2
2µ+p2
2µ
)), (3.79)
if it depends on both q and p. The short-hand notation in Eq. (3.73) can thus be extended viathe definition
QijkL (q, p) := lim
ε→0QijkL (q, p; 0) = QL
(mi
qp
(q2
2µij+
p2
2µjk
)). (3.80)
In the inhomogeneous term where the Legendre function only depends on p and k p one findsin the same way
limε→0
m
kpQL
(m
kp
(k2
2µ′+p2
2µ− E
)− iε
)pE,k−→ lim
ε→0
m
kpQL
(m
kp
(p2
2µ
)− iε
)
=m
kpQL
(m
2µ
p
k
). (3.81)
In Ref. [165] one can find a formal argument that one can neglect the contribution of the inhomo-geneous term compared to the the homogeneous part. As a less formal but more straightforwardmethod one can instead use that QL(z) can be written in terms of a hypergeometric function
2F1 which is proportional to∑∞
n=0 z−(2n+L+1) [164, 165]. Thus, Eq. (3.81) has the following
proportionality:
m
kpQL
(m
2µ
p
k
)∼ m
kp
∞∑
n=0
(m
2µ
p
k
)−(2n+L+1)
= m∞∑
n=0
(2µ
m
)(2n+L+1)k(2n+L)
p(2n+L+2)
pk−→ m∞∑
n=0
(2µ
m
)(2n+L+1)1
p(2n+L+2) 1 . (3.82)
Consequently, in our further analysis we can for asymptotic large off-shell momenta neglect theinhomogeneous terms in Eqs. (F.4 - F.9) and replace the denominators of the integrands usingEq. (3.77) and the Legendre functions via Eq. (3.80). Due to the vanishing inhomogeneities allamplitudes become independent of k and since E does not appear anymore, too, we simply writeT
(′)ij (p) ∼
∫dqT
(′)ij (q) in the following. Since p is unequal to zero one can additionally define new
amplitudes according to
T(′)ij (p) := p T
(′)ij (p) , ∀ i < j ∈ 1, 2, 3 , (3.83)
in order to find Eqs. (F.10 - F.15) written in terms of these new amplitudes and shown inappendix F.
48
3.4.1 Scale and inversion invariance and decoupling of amplitudes
The coupled integral equation system of Eqs. (F.10 - F.15) is scale invariant, but not invariantunder the inversion q → 1/q for large loop momenta. The reason for this is that in each termQL depends on the momentum q, but with different prefactors in front of q and in front of q−1.The prefactors themselves are not problematic. As long as in each QL, q and q−1 have the sameone, the equation system is inversion invariant, but if only one of them is different in at least oneQL the invariance is destroyed. However, we know from Refs. [92, 167] that this is an essentialproperty of a three particle system if one wants to decouple the integral equation system. Froma mathematical point of view the necessity of inversion invariance can be motivated by the factthat one can factor out the QL’s (which are then equal in each term), write the equations systemas a matrix equation, decouple the amplitudes and insert power law solutions for these decoupledamplitudes (a power law is the most straightforward choice for a scale and inversion invariantand physically meaningful amplitude [80]). After these steps one can use a Mellin transformto calculate the dq integral which leads to transcendental equations for the angular momentumdependent exponent of the power law s
(L)i (one equation per amplitude i). These equations have
purely complex solutions, that is, the Efimov effect is present, if a parameter λ (which dependson spin / isospin factors, symmetry factors, etc.) is larger than a critical value. To apply thismethod to our coupled integral equation system and to discuss the related issues in more detailwill be the main task for the rest of this section.
We deduce from Eqs. (F.10 - F.15) that the problematic prefactors mentioned above are causedby the fact that in general the masses m1, m2 and m3 are different. Thus, only for (approxi-mately) equal masses the equation system is (approximately) inversion invariant and a (so-calledapproximate) Efimov effect could be present in the system. However, if there is not a dimer state
for all combinations of particles (e.g. no d23 and thus no T(′)23 ) it may exist a scale and inversion
invariant equation subsystem with different masses. To be more precisely a not-existing dimerstate means that there is no resonant interaction with large scattering length between the twocorresponding particles. Nevertheless, they could have a deeply bound two-body state which isnot described by EFT(/π).As a measure for the mass difference between particles Pi and Pj we define a parameter
εij :=mi −mj
mi +mj
∀ i, j ∈ 1, 2, 3 , (3.84)
with the following properties:
1− εij =2mj
mi +mj
, (3.85)
1 + εij =2mi
mi +mj
, (3.86)
mi
µij=
2
1− εij, (3.87)
mj
µij=
2
1 + εij, (3.88)
49
√µijµ(ij)k
=
√1− εij1− εik
− 1
4(1− εij)2 . (3.89)
It can be used to eliminate one mass parameter from the coupled integral equations via
mj =1 + εij1− εij
mi . (3.90)
To ensure at least approximate scale and inversion invariance we will see that it is necessaryto have εij 1 so that one can neglect it and write mj ≈ mi. The error due to this re-placement depends on the exact value of εij. However, we will assume that it is at most ofthe order of the error coming from effective range corrections which are anyway neglected inour derivation of the coupled integral equations. Which mass can be replaced depends on theproperties of the specific system. In fact, one can identify three possible types of systems:
System types
• Type 1: The full system with shallow P1P2, P1P3 and P2P3 bound or virtual states.
• Type 2: The subsystem with only P1P2 and P1P3 bound or virtual states. Note, thatthe subsystem with only P1P2 and P2P3 bound states instead can be transformed intothe first one by interchanging the particle allocation of P1 and P2 which does not changethe dimer d12 in a physical sense. Thus, it is enough to consider the first subsystem.
• Type 3: The subsystem with only P1P2 bound or virtual states (i.e. d12 and d′12). Note,that one allocates the three particles in the system so that the scattering one is interestedin is always between particle P1 and P2. Since in our framework the scattering of twoparticles is only possible if there exists a dimer state (i.e. a two-body state with largescattering length) between these two particles, this subsystem is the only choice if thereare bound states only between two of the three particles.
At this point we have to make some important remarks: firstly, about not existing PiPj boundstates and secondly, regarding the counting scheme of identical diagrams explained in appendix C.In general one would transform the coupled integral equation system into a 6×6 matrix equationbecause we have started with six amplitudes. However, if there is no PiPj bound or virtual state
in the system one has to erase all terms proportional to T(′)(L)ij and thus to T
(′)(L)ij from the ana-
lysis since the corresponding Feynman diagrams do not exist in this case. Mathematically thisis allowed since the characteristic polynomial of a 6× 6 matrix with two rows and columns filledwith zeros is identical to the characteristic polynomial of the same matrix where one has erasedthe zero-filled rows and columns. If there is just one PiPj bound or virtual state, i.e. dij exists,
but d′ij not, then one erases all terms proportional to T′(L)ij and thus to T
′(L)ij from the analysis,
but keeps the unprimed amplitudes whose Feynman diagrams are still present. Consequently,there is just one row and column filled with zeros in the 6× 6 matrix which can thus be reducedto a 5× 5 matrix. The procedure how one deals with not existing dimers is thus clear. But whathappens in case of two identical particles? If Pi and Pj are identical it is clear that also the ampli-
tudes T(′)(L)ik and T
(′)(L)jk are mathematically and physically identical. Therefore one has to erase
50
the second amplitude T(′)(L)jk from the analysis since the just by hand inserted (cf. appendix C)
factors of (1 − δPiPj/2) in front of the Feynman diagrams are chosen exactly in the way that
one can also neglect these non-zero amplitudes. Consequently, it remain only T(′)(L)12 amplitudes
if all three particles are identical. However, this does not mean that one deals with a type 3system since there are bound states between e.g. P1 and P3, but they are equal to P1P2 boundstates. In summary we have the following rules (remember that a not present dimer meansthat there is no resonant interaction with large scattering length between the two particles):
Rules on how to treat not-existing dimers and identical particles in a system
• no dij and no d′ij dimer ⇒ erase all terms proportional to T(L)ij and T
′(L)ij .
• dij dimer present, but no d′ij dimer ⇒ erase all terms proportional to T′(L)ij .
• if Pi = Pj ⇒ keep amplitude T(′)(L)ik , but erase all terms proportional to T
(′)(L)jk .
The advantage of the less straightforward counting scheme of identical diagrams is that one alsoneeds less parameters which have to be calculated.
3.4.2 Type 1 systems
Consider the full system where bound or virtual states between all three particles exist. Namely,at least d12, d13 and d23 exist. To have a general expression we also assume the primed dimers tobe present and consequently no amplitude is erased from Eqs. (F.10 - F.15). As we have alreadynoticed these equations are not inversion invariant for q → 1/q. To achieve this property onehas to ensure that the argument of the Legendre function QL is equal in each term. Thus, it isrequired that all three masses are at least approximately equal in order to have a system wherethe Efimov effect could at all be present. This means that the mass difference parameters ε12,ε13 and ε23 must be small and from their definition in Eq. (3.84) follows that one can replace m3
and m2 via
m3 =1 + ε13
1− ε13
m1 ≈ m1 (for ε13 1) , (3.91)
m2 =1 + ε12
1− ε12
m1 ≈ m1 (for ε12 1) . (3.92)
Furthermore, the reduced masses µ13 and µ23 can be replaced by µ12, respectively, since
µ13 =m1m3
m1 +m3
=m1m2
1−ε231+ε23
m1 +m2
≈ µ12 (for ε23 1) , (3.93)
µ23 =m2m3
m2 +m3
=m1m2
1−ε131+ε13
m1 +m2
≈ µ12 (for ε13 1) . (3.94)
Applying these approximations to Eqs. (F.10 - F.15) one can factor out
∫ ΛC
0
dq
qQ122L (q, p) =
∫ ΛC
0
dq
qQL
(m1
qp
(q2
2µ12
+p2
2µ12
)),
51
and write the coupled integral equation system as a 6× 6 matrix equation:
T(L)12 (p)
T(L)13 (p)
T(L)23 (p)
T′(L)12 (p)
T′(L)13 (p)
T′(L)23 (p)
=(−1)L
π
m1
µ12
1√µ12µ(12)2
A1
∫ ΛC
0
dq
qQL
(m1
qp
(q2
2µ12
+p2
2µ12
))
T(L)12 (q)
T(L)13 (q)
T(L)23 (q)
T′(L)12 (q)
T′(L)13 (q)
T′(L)23 (q)
. (3.95)
Since m1 = m2 or at least m1 ≈ m2 one finds
µ12 ≈m1m1
m1 +m1
=m1
2, (3.96)
and thus
µ12
µ(12)2
≈ m1
2
(m1 +m1) +m1
(m1 +m1)m1
=m1
2
3m1
2m21
=3
4. (3.97)
Therefore the matrix equation simplifies to
T(L)12 (p)
T(L)13 (p)
T(L)23 (p)
T′(L)12 (p)
T′(L)13 (p)
T′(L)23 (p)
= (−1)L4√3 πA1
∫ ΛC
0
dq
qQL
(q
p+p
q
)
T(L)12 (q)
T(L)13 (q)
T(L)23 (q)
T′(L)12 (q)
T′(L)13 (q)
T′(L)23 (q)
, (3.98)
with A1 being a 6× 6 matrix whose columns are given by Eqs. (3.99 - 3.104).
52
(A1)i1 =
(1− δP1P2
2
)1
S12 c12
[x2 δ
(12)P1P3
f(3)(12)(12) S123 + x2 δ
(12)P2P3
f(3)(12)(12) S213
]
(1− δP1P2
2
)(y2 δP1P2 S123+y2 S213
)
√S13 S12 c13 c12
f(2)(12)(13)
(1− δP1P2
2
)(z2 S123+z2 δP1P2 S213
)
√S23 S12 c23 c12
f(1)(12)(23)
(1− δP1P2
2
)1
S12
√c′12 c12
[x′2 δ
(12)P1P3
f(3)(12)(12′) S123 + x′2 δ
(12)P2P3
f(3)(12)(12′) S213
]
(1− δP1P2
2
)(y′2 δP1P2 S123+y′2 S213
)
√S13 S12 c′13 c12
f(2)(12)(13′)
(1− δP1P2
2
)(z′2 S123+z′2 δP1P2 S213
)
√S23 S12 c′23 c12
f(1)(12)(23′)
, (3.99)
(A1)i2 =
(1− δP1P3
2
)(x3 δP1P3 S132+x3 S312
)
√S12 S13 c12 c13
f(3)(13)(12)
(1− δP1P3
2
)1
S13 c13
[y3 δ
(13)P1P2
f(1)(13)(13) S132 + y3 δ
(13)P2P3
f(3)(13)(13) S312
]
(1− δP1P3
2
)(z3 S132+z3 δP1P3 S312
)
√S23 S13 c23 c13
f(1)(13)(23)
(1− δP1P3
2
)(x′3 δP1P3 S132+x′3 S312
)
√S12 S13 c′12 c13
f(3)(13)(12′)
(1− δP1P3
2
)1
S13
√c′13 c13
[y′3 δ
(13)P1P2
f(1)(13)(13′) S132 + y′3 δ
(13)P2P3
f(3)(13)(13′) S312
]
(1− δP1P3
2
)(z′3 S132+z′3 δP1P3 S312
)
√S23 S13 c′23 c13
f(1)(13)(23′)
, (3.100)
53
(A1)i3 =
(1− δP2P3
2
)(x4 δP2P3 S231+x4 S321
)
√S12 S23 c12 c23
f(3)(23)(12)
(1− δP2P3
2
)(y4 S231+y4 δP2P3 S321
)
√S13 S23 c13 c23
f(2)(23)(13)
(1− δP2P3
2
)1
S23 c23
[z4 δ
(23)P1P2
f(2)(23)(23) S231 + z4 δ
(23)P1P3
f(3)(23)(23) S321
]
(1− δP2P3
2
)(x′4 δP2P3 S231+x′4 S321
)
√S12 S23 c′12 c23
f(3)(23)(12′)
(1− δP2P3
2
)(y′4 S231+y′4 δP2P3 S321
)
√S13 S23 c′13 c23
f(2)(23)(13′)
(1− δP2P3
2
)1
S23
√c′23 c23
[z′4 δ
(23)P1P2
f(2)(23)(23′) S231 + z′4 δ
(23)P1P3
f(3)(23)(23′) S321
]
, (3.101)
(A1)i4 =
(1− δP1P2
2
)1
S12
√c12 c′12
[x5 δ
(12′)P1P3
f(3)(12′)(12) S123 + x5 δ
(12′)P2P3
f(3)(12′)(12) S213
]
(1− δP1P2
2
)(y5 δP1P2 S123+y5 S213
)
√S13 S12 c13 c′12
f(2)(12′)(13)
(1− δP1P2
2
)(z5 S123+z5 δP1P2 S213
)
√S23 S12 c23 c′12
f(1)(12′)(23)
(1− δP1P2
2
)1
S12 c′12
[x′5 δ
(12′)P1P3
f(3)(12′)(12′) S123 + x′5 δ
(12′)P2P3
f(3)(12′)(12′) S213
]
(1− δP1P2
2
)(y′5 δP1P2 S123+y′5 S213
)
√S13 S12 c′13 c
′12
f(2)(12′)(13′)
(1− δP1P2
2
)(z′5 S123+z′5 δP1P2 S213
)
√S23 S12 c′23 c
′12
f(1)(12′)(23′)
, (3.102)
54
(A1)i5 =
(1− δP1P3
2
)(x6 δP1P3 S132+x6 S312
)
√S12 S13 c12 c′13
f(3)(13′)(12)
(1− δP1P3
2
)1
S13
√c13 c′13
[y6 δ
(13′)P1P2
f(1)(13′)(13) S132 + y6 δ
(13′)P2P3
f(3)(13′)(13) S312
]
(1− δP1P3
2
)(z6 S132+z6 δP1P3 S312
)
√S23 S13 c23 c′13
f(1)(13′)(23)
(1− δP1P3
2
)(x′6 δP1P3 S132+x′6 S312
)
√S12 S13 c′12 c
′13
f(3)(13′)(12′)
(1− δP1P3
2
)1
S13 c′13
[y′6 δ
(13′)P1P2
f(1)(13′)(13′) S132 + y′6 δ
(13′)P2P3
f(3)(13′)(13′) S312
]
(1− δP1P3
2
)(z′6 S132+z′6 δP1P3 S312
)
√S23 S13 c′23 c
′13
f(1)(13′)(23′)
, (3.103)
(A1)i6 =
(1− δP2P3
2
)(x7 δP2P3 S231+x7 S321
)
√S12 S23 c12 c′23
f(3)(23′)(12)
(1− δP2P3
2
)(y7 S231+y7 δP2P3 S321
)
√S13 S23 c13 c′23
f(2)(23′)(13)
(1− δP2P3
2
)1
S23
√c23 c′23
[z7 δ
(23′)P1P2
f(2)(23′)(23) S231 + z7 δ
(23′)P1P3
f(3)(23′)(23) S321
]
(1− δP2P3
2
)(x′7 δP2P3 S231+x′7 S321
)
√S12 S23 c′12 c
′23
f(3)(23′)(12′)
(1− δP2P3
2
)(y′7 S231+y′7 δP2P3 S321
)
√S13 S23 c′13 c
′23
f(2)(23′)(13′)
(1− δP2P3
2
)1
S23 c′23
[z′7 δ
(23′)P1P2
f(2)(23′)(23′) S231 + z′7 δ
(23′)P1P3
f(3)(23′)(23′) S321
]
. (3.104)
55
As a remainder we once more give the definition of all parameters appearing in the matrix A1:
• Pi := aiAi + biAi.
• δPiPj =
1 , if Pi = Pj
0 , elseand similarly for δAiAj .
• δ(ab(′))
PiPj:= δPiPj +
(δAiAj − δPiPj
)δη(′)ab |1|
(δη(′)ab |1|− δAaAb
),
with δη(′)ij |1|
:= δbaryon number 0 × δstrangeness 0 × δcharm 0 × δbeauty 0 × δtopness 0.
• f (h)
(ij(′))(kn[′]):=
aiajakan + v
(′)ij akan + ahw
(′)ij v
[′]kn + bibjv
[′]kn + aiajw
[′]kn + bhv
(′)ij w
[′]kn + w
(′)ij bkbn
aiajakan + v(′)ij akan + ahw
(′)ij w
[′]kn + bibjw
[′]kn + aiajv
[′]kn + bhv
(′)ij v
[′]kn + w
(′)ij bkbn
asdasdasdasd+ bibjbkbn , for i < k ∧ j = 2, k = 3 ∨ j = 3, k = 2+ bibjbkbn , else
.
• f (h)
(ij(′))(kn[′]):=
aiajakan + v
(′)ij bkbn + bhw
(′)ij w
[′]kn + aiajw
[′]kn + bibjv
[′]kn + ahv
(′)ij v
[′]kn + w
(′)ij akan
aiajakan + v(′)ij bkbn + bhw
(′)ij v
[′]kn + aiajv
[′]kn + bibjw
[′]kn + ahv
(′)ij w
[′]kn + w
(′)ij akan
asdasdasd+ bibjbkbn , for
i < k ∧ j = 2, k = 3 ∨ j = 3, k = 2
∨ i = k ∧ j = n
+ bibjbkbn , else.
• v(′)ij := δ
η(′)ij |1|
[1√2
+ δAiAj
(1− 1√
2
)](biaj + aibj) (η
(′)ij )δAiAj +
(1− δ
η(′)ij |1|
)biaj.
• w(′)ij := δ
η(′)ij |1|
[1√2
+ δAiAj
(1− 1√
2
)](biaj + aibj)
(1− δAiAj
)η
(′)ij .
• w(′)ij := w
(′)ij + δAiAj v
(′)ij .
• first symmetry factor Sij =
2 , if Pi = Pj
1 , else.
• second symmetry factor: Sijk = ζijk(1 + δPiPj + δPjPk + δPiPj δPjPk
),
56
with fermion minus sign ζijk:
ζijk =
−1 , if
• Pi 6= Pj 6= Pk ∧ only Pi, Pj fermions ∧i > j < k ∨ i > j > k
• Pi 6= Pj 6= Pk ∧ only Pj, Pk fermions ∧i < j > k ∨ i > j > k
• Pi 6= Pj 6= Pk ∧ Pi, Pj, Pk fermions ∧i > j < k ∨ i < j > k
• Pi = Pj 6= Pk ∧ Pi, Pj, Pk fermions ∧ i > k
• Pi = Pk 6= Pj ∧ Pi, Pj, Pk fermions .
• Pi 6= Pj = Pk ∧ only Pj fermion
• Pi 6= Pj = Pk ∧ Pi, Pj, Pk fermions ∧ i < j
• Pi = Pj = Pk ∧ Pi, Pj, Pk fermions
+1 , else
(3.105)
• c(′)ij is the normalization factor of the (iso)spin operators defined via(O(′)ij
)α,γσ
(O(′)†ij
)β,σγ
=
((O(′)ij
)α
(O(′)†ij
)β
)
γγ
= Tr
((O(′)ij
)α
(O(′)†ij
)β
)=: c
(′)ij δαβ.
Note, that all projectors given explicitly in appendix A are normalized which then impliesthat c
(′)ij = 1 ∀ i < j ∈ 1, 2, 3.
• x, y, z are spin and isospin dependent real numbers defined in Eqs. (A.40 - A.81) whichcan be calculated by hand using the tables in appendix A. A second method would beto use Mathematica and the 6-J symbol representation explained in the same appendix.Note, that using one of these two suggested methods implies c
(′)ij = 1 since both are based
on normalized projectors (see previous point).
In the next step we decouple the amplitudes: the matrix A1 can be diagonalized using an unitarytransformation matrix S which fulfills
S−1A1S = diag(λ1, λ2, λ3, λ4, λ5, λ6
), (3.106)
with λi (i ∈ 1, 2, 3, 4, 5, 6) being the eigenvalues of the matrix A1. Applying the transformationmatrix S to Eq. (3.98) and defining a new set of amplitudes,
T(L)1 (p)
T(L)2 (p)
T(L)3 (p)
T(L)4 (p)
T(L)5 (p)
T(L)6 (p)
:= S−1
T(L)12 (p)
T(L)13 (p)
T(L)23 (p)
T′(L)12 (p)
T′(L)13 (p)
T′(L)23 (p)
, (3.107)
57
one can write Eq. (3.98) in terms of these decoupled amplitudes:
T(L)1 (p)
T(L)2 (p)
T(L)3 (p)
T(L)4 (p)
T(L)5 (p)
T(L)6 (p)
= (−1)L4√3 π
diag(λ1, λ2, λ3, λ4, λ5, λ6
)∫ ΛC
0
dq
qQL
(q
p+p
q
)
T(L)1 (q)
T(L)2 (q)
T(L)3 (q)
T(L)4 (q)
T(L)5 (q)
T(L)6 (q)
.
(3.108)
Each row of this matrix equation leads to one (decoupled) integral equation for T(L)i (p) which is
in the limit of ΛC going to infinity scale invariant and symmetric under the inversion q → 1/q,i.e. inversion invariant. As explained in Refs. [80,92] the solutions to the integral equations thushave – for ΛC →∞ – the form of a power law:
T(L)i (p) = ps
(L)i , s
(L)i ∈ C ∀ i ∈ 1, 2, 3, 4, 5, 6 , (3.109)
which is the most straightforward and physically meaningful solution obeying the mentionedsymmetries. Replacing all amplitudes by their corresponding power law solution one finds
ps(L)1
ps(L)2
ps(L)3
ps(L)4
ps(L)5
ps(L)6
= (−1)L4√3 π
diag(λ1, λ2, λ3, λ4, λ5, λ6
)∫ ∞
0
dq
qQL
(q
p+p
q
)
qs(L)1
qs(L)2
qs(L)3
qs(L)4
qs(L)5
qs(L)6
. (3.110)
Note, that we have taken the limit ΛC →∞ which does not cause problems since the remainingintegrals are not divergent anymore. It is now useful to substitute q = Xp [165],
ps(L)1
ps(L)2
ps(L)3
ps(L)4
ps(L)5
ps(L)6
= (−1)L4√3 π
diag(λ1, λ2, λ3, λ4, λ5, λ6
)∫ ∞
0
dX
XQL
(X +
1
X
)
(Xp)s(L)1
(Xp)s(L)2
(Xp)s(L)3
(Xp)s(L)4
(Xp)s(L)5
(Xp)s(L)6
,
(3.111)
58
and – as the integral equations are decoupled – to divide the i-th row by ps(L)i to get
111111
= (−1)L4√3 π
diag(λ1, λ2, λ3, λ4, λ5, λ6
)∫ ∞
0
dX
Xs(L)1 −1
Xs(L)2 −1
Xs(L)3 −1
Xs(L)4 −1
Xs(L)5 −1
Xs(L)6 −1
QL
(X +
1
X
), (3.112)
which finally is equivalent to
1 = (−1)L4√3 π
λi
∫ ∞
0
dX Xs(L)i −1 QL
(X +
1
X
), for i = 1, 2, 3, 4, 5, 6 . (3.113)
To achieve our main goal, namely to verify whether the Efimov effect is present in a given threeparticle system or not, we need to check if the parameter s
(L)i is real or purely imaginary. Since
we know that only in the latter case it is present (see section 2.1 or Refs. [80,167] and references
therein). In order to derive the needed transcendental equation for each s(L)i one has to solve the
remaining integral in Eq. (3.113). One method would be to insert for arbitrary L the explicitform of QL. For L = 0 this is perfectly suitable (see for example Refs. [80,92,167]), but alreadyfor L = 1 one ends up with an integral which cannot be solved in an easy way. Therefore wefollow the work of Grießhammer [165] who derived a general expression for the integral
∫ ∞
0
dX Xs(L)i −1 QL
(X +
1
X
)
using hypergeometric functions and the Mellin transform. We use his derivation to find a solutionto the slightly more general integral
∫ ∞
0
dX Xs(L)i −1 QL
(αX + β
1
X
),
where the coefficients of q and 1/q are arbitrary (although we need α = β for the inversioninvariance). In appendix E we deduced in Eq. (E.18) the following result for the Mellin transformM of the Legendre function of the second kind with the mentioned structure of its argument:
M[QL
(αX + β
1
X
), s
(L)i
]:=
∫ ∞
0
dX Xs(L)i −1QL
(αX + β
1
X
)
=√π 2−(L+2)
(β
α
) s(L)i2(
1√αβ
)L+1 Γ
(L+s
(L)i +1
2
)Γ
(L−s(L)
i +1
2
)
Γ(
2L+32
)
× 2F1
(L+ s
(L)i + 1
2,L− s(L)
i + 1
2;2L+ 3
2;
1
4αβ
). (3.114)
59
Given in terms of the Γ-function and the hypergeometric function
pFq(a1, ..., ap; b1, ..., bq; z) =∞∑
k=0
p∏
i=1
Γ(k + ai)
Γ(ai)
q∏
j=1
Γ(bj)
Γ(k + bj)
zk
k!, p, q ∈ N . (3.115)
It is known that due to parity conservation only even or odd partial waves can mix. Togetherwith the argument that one considers low-energy scattering it is thus justified to assume S-wavescattering or – if L = 0 interactions are forbidden by some symmetry – D-wave scattering aloneand neglect all higher partial wave contributions. In appendix E it is shown that Eq. (3.114)yields for L = 0
∫ ∞
0
dX Xs(0)i −1Q0
(αX + β
1
X
)=
(√β
α
)s(0)i
π
s(0)i
sin(s
(0)i arcsin
(12
√1αβ
))
cos(π2s
(0)i
) , (3.116)
and for L = 1
∫ ∞
0
dX Xs(1)i −1Q1
(αX + β
1
X
)=
(√β
α
)s(1)i
πs(1)i[(
s(1)i
)2
− 1
] 1
sin(π2s
(1)i
)
×[√
4αβ − 1
s(1)i
sin
(s
(1)i arcsin
(1
2
√1
αβ
))
− cos
(s
(1)i arcsin
(1
2
√1
αβ
))]. (3.117)
Plugging these results into Eq. (3.113) one finds with α = β = 1 (since all particles have (roughly)
the same mass) for each parameter s(0)i a transcendental equation of the form
1 =4λi√
3
1
s(0)i
sin(π6s
(0)i
)
cos(π2s
(0)i
) , (3.118)
and for D-wave scattering a similar one for s(1)i :
1 =4 λi√
3
s(1)i[(
s(1)i
)2
− 1
]cos(π6s
(1)i
)
sin(π2s
(1)i
) − 4 λi1[(
s(1)i
)2
− 1
]sin(π6s
(1)i
)
sin(π2s
(1)i
) , (3.119)
where we have used that arcsin(1/2) = π/6.We note that Eq. (3.118) and Eq. (3.119) have exactly the same form as in the three spinlessboson case. This comes not as a surprise because all particles and dimers in a non-relativisticeffective theory are described in the same way. Hence, their momentum and energy dependence
60
and their behavior for asymptotic large momenta is independent of their spin, isospin, speciesand mass ratios. The latter would in principle influence the overall prefactor of the matrix A1,but as we must require all masses to be equal, even this prefactor is unchanged compared to thethree boson case. However, all other additional degrees of freedom affect the eigenvalue λi sincethey determine the entries of the matrix A1 (for three spinless boson λ1 ≡ λ = 2 is simply theelement of an 1-dimensional matrix). To answer the question whether or not the Efimov effectappears in a system of three particles P1, P2, P3 one just has to determine the eigenvalues ofthe matrix A1 whose elements can be read off from the properties of the three particles. Onlythe calculation of the spin and isospin dependent parameters x, y and z needs some effort if onedoes it by hand and not with Mathematica using the 6-J symbol notation.Before we apply this result to different three particle systems we discuss the two remaining typesof three particle systems.
3.4.3 Type 2 systems
In a system without shallow P2P3 bound or virtual states one has – according to the rules derivedat the beginning of this section (see page 51) – to erase all terms proportional to T
(L)23 and T
′(L)23
from Eqs. (F.10 - F.15). On the one hand a δ(12(′))13 = 1 factor in terms proportional to QL(∼ m2)
sets m1 = m3 and on the other hand δ(13(′))12 = 1 yields m1 = m2 in terms with QL(∼ m3). Due
to this fact it is sufficient to set m2 at least approximately equal to m3 in order to get a scaleand inversion invariant system necessary for the Efimov effect. What does this statement meanin few words? In fact, it means that we are able to analyze a system with two particles of equalmass and one particle with different mass where only two of the three pairs have a large scattinglength (i.e. a dimer state), if the two equal mass particles are the pair without dimer. Otherwise,it is not possible to decouple the amplitudes which is necessary for our method.If we indeed assume ε23 1 we can replace in the remaining equations all m3 and µ13 via
m3 =1 + ε23
1− ε23
m2 ≈ m2 ,
µ13 =m1m3
m1 +m3
=m1m2
1−ε231+ε23
m1 +m2
≈ µ12 . (3.120)
Together with the modified Kronecker-deltas this leads to a set of equations all proportional to
Q122L (q, p) = QL
(m1
qp
(q2
2µ12+ p2
2µ12
))which can be found in Eqs. (3.122 - 3.125). In the same
way as we did for type 1 systems one can now factor out the Legendre function QL and writeEqs. (3.122 - 3.125) as a 4× 4 matrix equation:
T(L)12 (p)
T(L)13 (p)
T′(L)12 (p)
T′(L)13 (p)
=
(−1)L
π
m1
µ12
1√µ12µ(12)2
A2
∫ ΛC
0
dq
qQL
(m1
qp
(q2
2µ12
+p2
2µ12
))
T(L)12 (q)
T(L)13 (q)
T′(L)12 (q)
T′(L)13 (q)
, (3.121)
where the matrix A2 has in principle the same entries as for a type 1 system, but with erasedthird and sixth row and column as it is shown in Eqs. (3.126 - 3.129).
61
T(L)12 (p) =
(−1)L(
1− δP1P2
2
)1
π
1
µ12 S12 c12
∫ ΛC
0
dq
q
T(L)12 (q)√µ12µ(12)2
×[x2 δ
(12)P1P3
f(3)(12)(12) S123 m1Q
122L (q, p) + x2 δ
(12)P2P3
f(3)(12)(12) S213 m1Q
122L (q, p)
]
+ (−1)L(
1− δP1P3
2
)1
π
(x3 δP1P3 S132 + x3 S312
)
µ12
√S12 S13 c12 c13
f(3)(13)(12)
∫ ΛC
0
dq
q
T(L)13 (q)√µ12µ(12)2
m1Q122L (q, p)
+ (−1)L(
1− δP1P2
2
)1
π
1
µ12 S12
√c12 c′12
∫ ΛC
0
dq
q
T′(L)12 (q)√
µ12µ(12)2
×[x5 δ
(12′)P1P3
f(3)(12′)(12) S123 m1Q
122L (q, p) + x5 δ
(12′)P2P3
f(3)(12′)(12) S213 m1Q
122L (q, p)
]
+ (−1)L(
1− δP1P3
2
)1
π
(x6 δP1P3 S132 + x6 S312
)
µ12
√S12 S13 c12 c′13
f(3)(13′)(12)
∫ ΛC
0
dq
q
T′(L)13 (q)√
µ12µ(12)2
m1Q122L (q, p) ,
(3.122)
T(L)13 (p) =
(−1)L(
1− δP1P2
2
)1
π
(y2 δP1P2 S123 + y2 S213
)
µ12
√S13 S12 c13 c12
f(2)(12)(13)
∫ ΛC
0
dq
q
T(L)12 (q)√µ12µ(12)2
m1Q122L (q, p)
+ (−1)L(
1− δP1P3
2
)1
π
1
µ12 S13 c13
∫ ΛC
0
dq
q
T(L)13 (q)√µ12µ(12)2
×[y3 δ
(13)P1P2
f(1)(13)(13) S132 m1Q
122L (q, p) + y3 δ
(13)P2P3
f(3)(13)(13) S312 m1Q
122L (q, p)
]
+ (−1)L(
1− δP1P2
2
)1
π
(y5 δP1P2 S123 + y5 S213
)
µ12
√S13 S12 c13 c′12
f(2)(12′)(13)
∫ ΛC
0
dq
q
T′(L)12 (q)√
µ12µ(12)2
m1Q122L (q, p)
+ (−1)L(
1− δP1P3
2
)1
π
1
µ12 S13
√c13 c′13
∫ ΛC
0
dq
q
T′(L)13 (q)√
µ12µ(12)2
×[y6 δ
(13′)P1P2
f(1)(13′)(13) S132 m1Q
122L (q, p) + y6 δ
(13′)P2P3
f(3)(13′)(13) S312 m1Q
122L (q, p)
], (3.123)
62
T′(L)12 (p) =
(−1)L(
1− δP1P2
2
)1
π
1
µ12 S12
√c′12 c12
∫ ΛC
0
dq
q
T(L)12 (q)√µ12µ(12)2
×[x′2 δ
(12)P1P3
f(3)(12)(12′) S123 m1Q
122L (q, p) + x′2 δ
(12)P2P3
f(3)(12)(12′) S213 m1Q
122L (q, p)
]
+ (−1)L(
1− δP1P3
2
)1
π
(x′3 δP1P3 S132 + x′3 S312
)
µ12
√S12 S13 c′12 c13
f(3)(13)(12′)
∫ ΛC
0
dq
q
T(L)13 (q)√µ12µ(12)2
m1Q122L (q, p)
+ (−1)L(
1− δP1P2
2
)1
π
1
µ12 S12 c′12
∫ ΛC
0
dq
q
T′(L)12 (q)√
µ12µ(12)2
×[x′5 δ
(12′)P1P3
f(3)(12′)(23′) S123 m1Q
122L (q, p) + x′5 δ
(12′)P2P3
f(3)(12′)(12′) S213 m1Q
122L (q, p)
]
+ (−1)L(
1− δP1P3
2
)1
π
(x′6 δP1P3 S132 + x′6 S312
)
µ12
√S12 S13 c′12 c
′13
f(3)(13′)(12′)
∫ ΛC
0
dq
q
T′(L)13 (q)√
µ12µ(12)2
m1Q122L (q, p) ,
(3.124)
T′(L)13 (p) =
(−1)L(
1− δP1P2
2
)1
π
(y′2 δP1P2 S123 + y′2 S213
)
µ12
√S13 S12 c′13 c12
f(2)(12)(13′)
∫ ΛC
0
dq
q
T(L)12 (q)√µ12µ(12)2
m1Q122L (q, p)
+ (−1)L(
1− δP1P3
2
)1
π
1
µ12 S13
√c′13 c13
∫ ΛC
0
dq
q
T(L)13 (q)√µ12µ(12)2
×[y′3 δ
(13)P1P2
f(1)(13)(13′) S132 m1Q
122L (q, p) + y′3 δ
(13)P2P3
f(3)(13)(13′) S312 m1Q
122L (q, p)
]
+ (−1)L(
1− δP1P2
2
)1
π
(y′5 δP1P2 S123 + y′5 S213
)
µ12
√S13 S12 c′13 c
′12
f(2)(12′)(13′)
∫ ΛC
0
dq
q
T′(L)12 (q)√
µ12µ(12)2
m1Q122L (q, p)
+ (−1)L(
1− δP1P3
2
)1
π
1
µ12 S13 c′13
∫ ΛC
0
dq
q
T′(L)13 (q)√
µ12µ(12)2
×[y′6 δ
(13′)P1P2
f(1)(13′)(13′) S132 m1Q
122L (q, p) + y′6 δ
(13′)P2P3
f(3)(13′)(13′) S312 m1Q
122L (q, p)
]. (3.125)
63
(A2)i1 =
(1− δP1P2
2
)1
S12 c12
[x2 δ
(12)P1P3
f(3)(12)(12) S123 + x2 δ
(12)P2P3
f(3)(12)(12) S213
]
(1− δP1P2
2
)(y2 δP1P2 S123+y2 S213
)
√S13 S12 c13 c12
f(2)(12)(13)
(1− δP1P2
2
)1
S12
√c′12 c12
[x′2 δ
(12)P1P3
f(3)(12)(12′) S123 + x′2 δ
(12)P2P3
f(3)(12)(12′) S213
]
(1− δP1P2
2
)(y′2 δP1P2 S123+y′2 S213
)
√S13 S12 c′13 c12
f(2)(12)(13′)
, (3.126)
(A2)i2 =
(1− δP1P3
2
)(x3 δP1P3 S132+x3 S312
)
√S12 S13 c12 c13
f(3)(13)(12)
(1− δP1P3
2
)1
S13 c13
[y3 δ
(13)P1P2
f(1)(13)(13) S132 + y3 δ
(13)P2P3
f(3)(13)(13) S312
]
(1− δP1P3
2
)(x′3 δP1P3 S132+x′3 S312
)
√S12 S13 c′12 c13
f(3)(13)(12′)
(1− δP1P3
2
)1
S13
√c′13 c13
[y′3 δ
(13)P1P2
f(1)(13)(13′) S132 + y′3 δ
(13)P2P3
f(3)(13)(13′) S312
]
, (3.127)
(A2)i3 =
(1− δP1P2
2
)1
S12
√c12 c′12
[x5 δ
(12′)P1P3
f(3)(12′)(12) S123 + x5 δ
(12′)P2P3
f(3)(12′)(12) S213
]
(1− δP1P2
2
)(y5 δP1P2 S123+y5 S213
)
√S13 S12 c13 c′12
f(2)(12′)(13)
(1− δP1P2
2
)1
S12 c′12
[x′5 δ
(12′)P1P3
f(3)(12′)(12′) S123 + x′5 δ
(12′)P2P3
f(3)(12′)(12′) S213
]
(1− δP1P2
2
)(y′5 δP1P2 S123+y′5 S213
)
√S13 S12 c′13 c
′12
f(2)(12′)(13′)
, (3.128)
(A2)i4 =
(1− δP1P3
2
)(x6 δP1P3 S132+x6 S312
)
√S12 S13 c12 c′13
f(3)(13′)(12)
(1− δP1P3
2
)1
S13
√c13 c′13
[y6 δ
(13′)P1P2
f(1)(13′)(13) S132 + y6 δ
(13′)P2P3
f(3)(13′)(13) S312
]
(1− δP1P3
2
)(x′6 δP1P3 S132+x′6 S312
)
√S12 S13 c′12 c
′13
f(3)(13′)(12′)
(1− δP1P3
2
)1
S13 c′13
[y′6 δ
(13′)P1P2
f(1)(13′)(13′) S132 + y′6 δ
(13′)P2P3
f(3)(13′)(13′) S312
]
, (3.129)
64
Although the matrices A1 and A2 are almost identical (they just differ by their dimension, butnot by their entries) Eq. (3.98) and Eq. (3.121) have a different, mass dependent prefactor,
respectively, which can affect the result for s(L)i in the end.
To find the corresponding transcendental equation for the parameter s(L)i we do the same as in the
type 1 case. One applies an unitary transformation under which the amplitudes decouple, namely,under which A2 becomes diagonal. Since the decoupled amplitudes fulfill scale and inversion
invariant integral equations their replacement by a power law solution ∼ ps(L)i (i ∈ 1, 2, 3, 4) is
justified and hence one finds in the limit ΛC →∞ a similar equation to Eq. (3.113):
1 =(−1)L
π
m1
µ12
1√µ12µ(12)2
λi
∫ ∞
0
dX Xs(L)i −1 QL
(m1
2µ12
(X +
1
X
)), (3.130)
where λi are now the eigenvalues of the matrix A2. Considering the argument of the Legendrefunction it becomes clear why we had to find the Mellin transform Eq. (3.114) of the more generalQL. Using Eq. (3.116) one finds with α = β = m1/(2µ12) the corresponding transcendentalequation for L = 0
1 =m1
µ12
õ(12)2
µ12
λi1
s(0)i
sin(s
(0)i arcsin
(µ12m1
))
cos(π2s
(0)i
) (3.131)
⇐⇒ 1 =4 λi√
4− [(1− ε12)2 − 2]2
1
s(0)i
sin(s
(0)i arcsin
((1−ε12)
2
))
cos(π2s
(0)i
) . (3.132)
In the second representation (Eq. (3.132)) we have with Eq. (3.89) and
4
x√
4− x2=
4√4x2 − x4
=4√
−(−4x2 + x4)=
4√− [(x2 − 2)2 − 4]
=4√
4− (x2 − 2)2, (3.133)
rewritten it in terms of the mass difference parameter −1 < ε12 < 1 which must not tend to zeroin a system of type 2. However, if it is very close to zero one directly observers that Eq. (3.132)
looks the same as the determining equation of s(0)i for a type 1 system (Eq. (3.118)), but as the
eigenvalues λi can be different for A1 and A2 they lead in general to different results for s(0)i . Only
in the case of a type 1 system without a P2P3 dimer where one has to erase all T(′)23 amplitudes
so that A1 becomes equal to A2 both equations are equivalent. But this is not surprising sincesuch a system (type 1 without d
(′)23) is what we have called type 2. Nevertheless, in general where
ε12 is not close to zero it can have a large effect on s(0)i : if one mass is much larger than the other
it tends to ±1. In the limit of ε12 → +1 the right-hand-side of Eq. (3.132) yields 0/0 and henceone can use the rule of L’Hospital and arcsin(0) = 0 to find
1 = limε12→+1
4 λi√4− [(1− ε12)2 − 2]2
1
s(0)i
sin(s
(0)i arcsin
((1−ε12)
2
))
cos(π2s
(0)i
)
= − λi
cos(π2s
(0)i
) , (3.134)
65
which can be solved for s(0)i :
s(0)i =
2
πarccos (−λi) =
2
π[π − arccos (λi)]
= 2 +2i
πln
(λi + i
√1− λ2
i
), (3.135)
where we have used the complex logarithm to continue the arc-cosine to the region outside theinterval [−1, 1]. For eigenvalues |λi| > 1, s
(0)i can be purely imaginary and hence the existence of
the Efimov effect is not excluded and one cannot make a general statement about its occurrence.In the other limit, i.e. ε12 → −1, Eq. (3.132) goes to
1 = limε12→−1
4 λi√4− [(1− ε12)2 − 2]2
1
s(0)i
sin(s
(0)i arcsin
((1−ε12)
2
))
cos(π2s
(0)i
)
=λi0
tan(π2s(0))
s(0)i
, (3.136)
with arcsin(x)x→1−→ π/2. Thus, s
(0)i tends to infinity, but its sign and in particular the fact
whether it is real or purely imaginary depends on the sign of λi. Using the definition of ε12 =(m1−m2)/(m1 +m2) we conclude that on the one hand the limit of it going to +1 is equivalentto m1 m2 = m3 and on the other hand ε12 → −1 corresponds to m1 m2 = m3. Theconclusion is thus that one cannot deduce solely from the mass ratio of the particles if there isan Efimov effect in the system or not.For completeness we state also the result of Eq. (3.130) for angular momentum L = 1 (cf.Eq. (3.117)):
1 = − m1
µ12
õ(12)2
µ12
λis
(1)i[(
s(1)i
)2
− 1
] 1
sin(π2s
(1)i
)
×
√(m1
µ12
)2
− 1
s(1)i
sin
(s
(1)i arcsin
(µ12
m1
))− cos
(s
(1)i arcsin
(µ12
m1
))
⇐⇒ 1 =4 λi√
4− [(1− ε12)2 − 2]2
s(1)i[(
s(1)i
)2
− 1
]cos(s
(1)i arcsin
((1−ε12)
2
))
sin(π2s
(1)i
)
− 4 λi(1− ε12)2
1[(s
(1)i
)2
− 1
]sin(s
(1)i arcsin
((1−ε12)
2
))
sin(π2s
(1)i
) . (3.137)
Taking the limit ε12 → ±1 does not lead to further insights regarding the Efimov effect. Hence,we omit its derivation. However, besides these subtleties the search for a possible Efimov effect
66
is like in the type 1 system reduced to the determination of eigenvalues of a matrix whose entriescan straightforwardly be read off from the properties of the three particles in the system.
Before we continue with the last type of systems we can – as a check – apply the type 2 resultsabove to a system of either two (identical or distinguishable) bosons or two identical fermionsand a third boson in order to reproduce (in parts) Fig. (53) in the work of Hammer and Braatenin Ref. [80]. To do so one has to change the particle allocation via P1 = PHB
3 , P2 = PHB2 and
P3 = PHB1 since in Ref. [80] a system with mHB
1 = mHB2 6= mHB
3 is considered which does not fitin our scheme with m1 6= m2 = m3. Assuming the bosons to be scalars and the fermions to havespin 1/2 one has the following dimers in the system: d12 and d13. Both with spin 1/2 (there areno primed dimers) so that the spin 0 scattering channel is the only allowed channel for bosonicand fermionic configurations. From the particle allocation we read off that S12 = S13 = 1, δPiPj =
δ(ab)PiPj
= 0 for i = 1 ∧ j = 2, 3 and that f(3)(12)(12) = f
(3)(12)(12) = f
(2)(12)(13) = f
(2)(12)(13) = 1. Furthermore,
we know from the symmetry factor Sijk derived in appendix B and given in Eq. (B.13) that forall above mentioned combinations of bosons and fermions it holds S213 = S312 = 1. The matrixA2 (cf. its first two columns Eq. (3.126) and Eq. (3.127)) simplifies to
A2 =
(x2 δ
(12)P2P3
x3
y2 y3 δ(13)P2P3
), (3.138)
which is a 2× 2 matrix since we erased all primed variables because there are no primed dimers.All in all there are three cases left:
• P2 6= P3 bosons: the modified Kronecker-deltas both vanish. Thus, one finds
A2 =
(0 x3
y2 0
)=
(0 11 0
),
where the parameters x3 and y2 are calculated using one of the methods explained in thelast two sections of appendix A. The matrix A2 has the eigenvalues λ1,2 = ±1.
• P2 = P3 bosons: the modified Kronecker-deltas yield 1 and in addition it holds T13 = T12
since P2 = P3. Hence, according to the rules in the box on page 51 one erases T13 from theequations. The result is a number A2 = x2 = 1 so that λ1 = 1.
• P2 = P3 fermions: the modified Kronecker-delta yield 1 and as above the matrix A2 againhas a 1× 1 structure, but now the x parameter is −1 so that one finds: A2 = x2 = −1 andthus λ1 = −1.
The non-positive eigenvalues in the fermionic case are already a hint to what is shown in Ref. [172]and used in a similar framework in Ref. [173]. Namely, that in a system as above the Efimoveffect can only occur in S-wave scattering if the particles are (identical or distinguishable) bosonsor in P -wave scattering if at least two of the particles are fermions. Indeed, one does not findpurely imaginary solutions to Eq. (3.131) for λ1 = −1, independently of the mass factors. Incontrast, for λ1 = 1 the same equation (3.131) yields purely imaginary solutions for arbitrary
67
0.01 0.1 1 10 100m
1/m
3
101
103
105
exp
(π/s
1
(0,1
) )
Figure 3.7: Results of type-2-system calculations for two either identical or distinguishablebosons (pluses) or identical fermions (crosses) P2, P3 and a third boson P1 in comparison with
Fig. (53) in [80] where the solid line describes the scaling factor exp(π/s(0)1 ) for P2, P3 being
bosons and the dash-dotted line that for P2, P3 being identical fermions, i.e. exp(π/s(1)1 ), as
functions of the mass ratio m1/m3.
mass ratios. Rewriting the mass dependent factors in Eq. (3.131) in terms of m1/m3 = m1/m2
one can reproduce the solid line of Fig. (53) in Ref. [80] which belongs to the two bosonic cases:we took this figure and used our method to find the ”pluses“ shown in Fig. 3.7 which matchwith the solid line as they should. In the same way one can reformulate the P -wave relationEq. (3.137) in terms of the mass ratio m1/m3 and search for purely imaginary solutions of s
(1)1 for
λ1 = −1 in the fermionic case. One finds that for mass ratios m1/m3 & 13.6 such solutions existand thus the Efimov effect is present, which agrees with the statements in Refs. [80, 172, 173].In Fig. 3.7 we have marked the results of our method by crosses which match with the dash-dotted line for fermions as expected. Note, that the dashed line of Fig. (53) in Ref. [80] whichcorresponds to a similar (though different) system of two bosons / fermions and a third boson,but with a bound state d23 between the particles with equal masses, cannot be described by ourmethod (see opening words of this section 3.4.3) and is thus not shown in Fig. 3.7.
3.4.4 Type 3 systems
We will now focus on three particle systems of type 3 where only shallow P1P2 dimers exist.As there is just one pair of particles with large scattering length one would expect that Efimovphysics are not relevant in such a system. Indeed, we will show that the analysis of type 3systems only yields an Efimov effect if the system can fluctuate into a type 2 system due to asuperposition in the flavor wave function of the respective dimer.Following the rules on page 51 one has to erase in a type 3 system all terms proportional to
68
T(′)(L)13 and T
(′)(L)23 from Eqs. (F.10 - F.15). It remain just two coupled integral equations:
T(L)12 (p) = (−1)L
(1− δP1P2
2
)1
π
1
µ12 S12 c12
∫ ∞
0
dq
q
T(L)12 (q)√µ12µ(12)3
×[x2 δ
(12)P1P3
f(3)(12)(12) S123 m2Q
211L (q, p) + x2 δ
(12)P2P3
f(3)(12)(12) S213 m1Q
122L (q, p)
]
+ (−1)L(
1− δP1P2
2
)1
π
1
µ12 S12
√c12 c′12
∫ ∞
0
dq
q
T′(L)12 (q)√
µ12µ(12)3
×[x5 δ
(12′)P1P3
f(3)(12′)(12) S123 m2Q
211L (q, p) + x5 δ
(12′)P2P3
f(3)(12′)(12) S213 m1Q
122L (q, p)
],
(3.139)
T′(L)12 (p) = (−1)L
(1− δP1P2
2
)1
π
1
µ12 S12
√c′12 c12
∫ ∞
0
dq
q
T(L)12 (q)√µ12µ(12)3
×[x′2 δ
(12)P1P3
f(3)(12)(12′) S123 m2Q
211L (q, p) + x′2 δ
(12)P2P3
f(3)(12)(12′) S213 m1Q
122L (q, p)
]
+ (−1)L(
1− δP1P2
2
)1
π
1
µ12 S12 c′12
∫ ∞
0
dq
q
T′(L)12 (q)√
µ12µ(12)3
×[x′5 δ
(12′)P1P3
f(3)(12′)(23′) S123 m2Q
211L (q, p) + x′5 δ
(12′)P2P3
f(3)(12′)(12′) S213 m1Q
122L (q, p)
].
(3.140)
We note that each term is proportional either to δ(12(′))P1P3
or to δ(12(′))P2P3
, but if P1 = P3 (P2 = P3)there must be an extra P2P3 (P1P3) bound state which is a contradiction to the classificationof type 3 systems. Thus, we conclude that a type 3 system only exists for the special case (cf.Eq. (C.2))
δ(12(′))P1/P2,P3
P1/P2 6=P3= δA1/A2,A3 δη(′)12 |1|
(δη(′)12 |1|− δA1A2
)= 1 . (3.141)
Furthermore, it also leads to a contradiction if we assume δ(12(′))P1P3
= δ(12(′))P2P3
= 1 because this wouldimply A1 = A2 = A3, but with A1 = A2 the equation above always yields zero. Therefore one has
to distinguish two cases for which a type 3 system can exist: firstly, δ(12(′))P1P3
= 1 and δ(12(′))P2P3
= 0.
And secondly, the vice versa case δ(12(′))P1P3
= 0 and δ(12(′))P2P3
= 1. However, by interchanging theparticle allocation of P1 and P2 one can transform these two cases into each other and we onlyhave to consider one of them. We chose the first one.
69
Case 1: δ(12(′))P1P3
= 1 and δ(12(′))P2P3
= 0
From the condition δ(12(′))P1P3
= 1 for P1 6= P3 it follows that δη(′)12 |1|
= 1, δA1A2 = 0 and δA1A3 = 1.
Therefore one can conclude that P1 and P2 both are bosons which implies ζ123 = +1 and thusS123 = 1 as well as S12 = 1. Also concerning the a, b parameters one finds some constraints:a1a2 = b1b2 = 0, v
(′)12 = w
(′)12 = 1/
√2 and a3 = 1 or b3 = 1. Altogether this leads to
f(3)
(12(′))(12[′])=(a1a2a1a2 + a3w
(′)12w
[′]12 + b3v
(′)12v
[′]12 + b1b2b1b2
)=
1
2, (3.142)
for all combinations of primed and unprimed variables. Finally, this yields with δP1P2 = 0 (followsfrom δA1A2 = 0) the matrix equation below:
(T
(L)12 (p)
T′(L)12 (p)
)=
(−1)L
π
m2
µ12
õ(12)1
µ12
x2c12
x5√c12c′12
x′2√c12c′12
x′5c′12
∫ ΛC
0
dq
qQL
(m2
2µ12
(q
p+p
q
))(T
(L)12 (q)
T′(L)12 (q)
).
(3.143)
Decoupling the amplitudes and inserting the power-law solution ∼ ps(L)i one finds in the limit
ΛC →∞ with X = q/p:
1 =(−1)L
π
m2
µ12
õ(12)1
µ12
λi
∫ ∞
0
dX
X
s(L)i −1
QL
(m2
2µ12
(X +
1
X
)), for i ∈ 1, 2 , (3.144)
with λi being the eigenvalues of the matrix
A3a =
x2c12
x5√c12c′12
x′2√c12c′12
x′5c′12
. (3.145)
For S-wave scattering we deduce from Eq. (3.116) with α = β = m2/(2µ12) the following trans-
cendental equation for s(0)i :
1 =m2
µ12
õ(12)1
µ12
λi1
s(0)i
sin(s
(0)i arcsin
(µ12m2
))
cos(π2s
(0)i
) . (3.146)
This result is very similar to what we have found for the type 2 system. However, this isnot surprising because a system of type 3 only exists in a situation where one deals with asuperposition-dimer d
(′)12 ∼ A2A1 + η
(′)12A2A1 (cf. the properties of P1 and P2 derived above). In
fact, it accounts for the case that one considers a three particle system A2A1A3 where neitherA1A3 nor A2A3 have a bound or virtual state and which seems to not contribute at all. But asit can fluctuate into the system A2A1A3 where the latter two fields have a dimer state, it doescontribute. Then, we finally note that the fluctuated system is because of A1 = A3 nothing elsethen a type 2 system with particles 1 and 2 interchanged compared to what we have discussesin section 3.4.3. This explains the mentioned similarity and we conclude that type 3 systems arejust a remnant of type 2 ones. Hence, they can be ignored in further analyses.
70
3.4.5 Implementation with Mathematica
Considering the matrices A1 and A2 corresponding to type 1 and type 2 three particle systemswe observe that they only depend on a rather small number of parameters like spin and isospin.Hence, it is relatively straightforward to implement the full calculation of the two mentionedmatrices into a computer program. For this purpose we choose Mathematica since it alreadycontains a command SixJSymbol which determines the Wigner 6-J symbols for given spins.Furthermore, one can – after determining the eigenvalues of A1,2 – use the FindRoot commandto find a solution of the transcendental equation for the scaling parameter s which dependson these eigenvalues. To use such a program in the right way one has, however, to rememberthe rules in the box on page 51 in order to assign to the parameters the right input values.Considering for example the NNN system one has to erase the amplitudes T13, T23 and therespective primed counterparts. Consequently, one has to set the spins J13 = J23 = no (see com-ments within the program) to get the right results. The corresponding Mathematica notebook”Efimov Analysis.nb“ can be found on the attached CD. On start the input parameters areset to values which correspond to the NNΛ system with the particle allocation P1 = P2 = N ,P3 = Λ and with the hypertriton as Efimov trimer.
71
Chapter 4
Efimov effect in hadronic molecules
After the discussion of the different types of three particle systems in the previous section onecan continue with two possible applications. Firstly, one can apply the derived methods forthe determination of the parameter s(L) to various three particle systems where some particlescatters off a – more or less – established hadronic molecule, in order to check if the Efimoveffect is present in such a system. The nature of many states appearing below is not completelyunderstood due to lacking experimental data. We will interpret all of them as S-wave hadronicmolecules. Additionally, we assume that they at least approximately fulfill the conditions that arenecessary to treat them in a pionless EFT (see also section 4.3). The second application will bethat we consider specific combinations of charm and bottom mesons (which are the constituentsof a large number of molecules, cf. Tab. 4.1) and check which spin and isospin configurationslead to Efimov trimers in the scattering off a third charm or bottom meson. This procedure isalso applied to hypothetical dibaryon systems.Before we start with this task we want to emphasize that the ideas of treating hadronic moleculeslike the X(3872) in EFT(/π) [132] or to search for possible Efimov trimers inX(3872)–D scattering[133] were already invented by Hammer and Braaten and respective co-workers. Moreover, oneshould also mention the work regarding hadronic molecules done in Refs. [134–162] which is inmany aspects the basis for this thesis.
4.1 Established systems
We start with the first application where we discuss the Efimov effect in established moleculesystems. For this purpose we have collected a set of experimentally known bound or virtualstates in Tab. 4.1 which have in common that one can interpret them as hadronic molecules.Furthermore, one can find the constituent particles of these molecules in the same table wherethe given mass of an isospin multiplet has to be understood as the charged/uncharged meanmass. A short discussion of the applicability of EFT(/π) to the particles in this table can befound in section 4.3. Before we discus the results of our search for Efimov trimers we start withtwo demonstrative examples to clarify the derived method: on the one hand the well-known threenucleon system with the triton as three-body bound state (discussed e.g. in Refs. [80, 92, 93])and on the other hand we consider as a ”new“ system the one made of three kaons.
72
particle / molecule wave function IG(JP) isospin av. mass [MeV]
K [2] 12(0−) 495.6
D [2] 12(0−) 1867.2
D∗ [2] 12(1−) 2008.6
D1 [2] 12(1+) 2421.4
B [2] 12(0−) 5279.4
B∗ [2] 12(1−) 5325.2
N [2] 12(1
2
+) 938.9
Λ [2] 0(12
+) 1115.7
a0(980) [2] KK [40–42,61] 1−(0+) 980
f0(980) [2] KK [40–42,61] 0+(0+) 990
D∗s0(2317) [36,37] DK [43, 44,47] 0(0+) 2317.8
Ds1(2460) [36,37] D∗K [43, 44,47] 0(1+) 2459.6
D∗s1(2700) [59] D1K (?) 0(1−) 2709
X(3872) [20] 1√2
(D∗D +D∗D
)[58] 0+(1+) 3871.7
Zc(3900) [22–24] 1√2
(D∗D +D∗D
)[49, 50] 1+(1+) 3899.0
Z1(4051) [32] −D∗D∗ [51, 52] 1−(1+) 4051
Z2(4250) [32] 1√2
(D1D −D1D
)[51, 53] 1−(1−) 4248
Z(4430) [29–31] 1√2
(D∗D1 +D∗D1
)[51, 54] 1+(1−) 4443
Zb(10610) [33] 1√2
(B∗B +B∗B
)[55] 1+(1+) 10608.4
Z ′b(10650) [33] B∗B∗ [55] 1+(1+) 10653.2
Λ(1405) [2] KN [56, 57] 0(12
−) 1405.1
3S1 NN PW (deuteron) NN 0(1+) 1875.61S0 NN PW NN 1(0−)
3S1 ΛN PW [38,39] ΛN 12(1+)
1S0 ΛN PW [38,39] ΛN 12(0−)
1S0 ΛΛ PW (?) ΛΛ [175] 0(0+)
Table 4.1: Considered hadrons and their two-body molecule states (PW: partial wave). Theisospin averaged mass is the mean one of the iso-multiplet members. Note, that most of themolecule wave functions are hypotheses (their first proposal is referenced); especially, the inter-pretation of D∗s1 = KD1 and the existence of the H-dibaryon ΛΛ are very hypothetical as indicatedby the question marks. Note also, the remarks on the applicability of EFT(/π) in section 4.3. Foreach particle the reference for its discovery and properties is given in the first column.
73
4.1.1 Three nucleon system
In the NN system it is known that there is one 3S1 partial wave bound state with isospin 0(the deuteron) and a 1S0 partial wave virtual state being an iso-triplet. Since the three particlesin the system are identical their allocation to the generic particles P1, P2 and P3 is obviouslyPi = N for i = 1, 2, 3. The deuteron will be assigned to the dimer d12 and the virtual 1S0 stateto d′12 with quantum numbers (cf. Tab. 4.1)
I(JP)
(d12) = 0(1+) ,
I(JP)
(d′12) = 1(0−) . (4.1)
Keeping in mind that nucleons have both spin and isospin given by 1/2 one concludes that oneonly has – according to the rules in the box on page 51 – to deal with T12 and T ′12. Since allparticles have the same mass we consider a type 1 system where the matrix A1 is a 2× 2 matrixwith elements
(A1)11 =
(1− δP1P2
2
)1
S12 c12
[x2 δ
(12)P1P3
f(3)(12)(12) S123 + x2 δ
(12)P2P3
f(3)(12)(12) S213
],
(A1)21 =
(1− δP1P2
2
)1
S12
√c′12 c12
[x′2 δ
(12)P1P3
f(3)(12)(12′) S123 + x′2 δ
(12)P2P3
f(3)(12)(12′) S213
],
(A1)12 =
(1− δP1P2
2
)1
S12
√c12 c′12
[x5 δ
(12′)P1P3
f(3)(12′)(12) S123 + x5 δ
(12′)P2P3
f(3)(12′)(12) S213
],
(A1)22 =
(1− δP1P2
2
)1
S12 c′12
[x′5 δ
(12′)P1P3
f(3)(12′)(12′) S123 + x′5 δ
(12′)P2P3
f(3)(12′)(12′) S213
].
The particle allocation yields that a1 = a2 = a3 = 1, but all b’s are zero and consequently thefunctions f and f return ”1“ independently of their arguments. Furthermore, all Kronecker-deltas are 1 and the symmetry factors S12 = 2 and S123 = S213 = −4 can be found accordingto appendix B. The remaining spin and isospin dependent factors x are determined using oneof the methods explained in appendix A. Depending on spin (S) and isospin (I) channel onefinds using for example the 6-J symbols in Eqs. (A.111 - A.146) with j1 = j2 = i1 = i2 = 1/2,J12 = I ′12 = 1 and J ′12 = I12 = 0 the following values:
x2 = x2 = (−1)2(J+I) × 3
12
12
112
S 1
× 1
12
12
012
I 0
=
−1
4, for S = 1
2& I = 1
212, for S = 3
2& I = 1
2
,
x′2 = x′2 = (−1)2(J+I) ×√
3
12
12
012
S 1
×√
3
12
12
112
I 0
=
34, for S = 1
2& I = 1
2
0 , for S = 32
& I = 12
,
x5 = x5 = (−1)2(J+I) ×√
3
12
12
112
S 0
×√
3
12
12
012
I 1
=
34, for S = 1
2& I = 1
2
0 , for S = 32
& I = 12
,
x′5 = x′5 = (−1)2(J+I) × 1
12
12
012
S 0
× 3
12
12
112
I 1
=
−1
4, for S = 1
2& I = 1
2
0 , for S = 32
& I = 12
.
74
All in all the matrix A1 reduces to
AS= 12,I= 1
21 =
(12−3
2
−32
12
), (4.2)
in the spin and isospin 1/2 channel (doublet channel) and to AS= 32,I= 1
21 = λ
S= 32,I= 1
21 = −1/2 in the
quartet channel with S = 32. The eigenvalues in the former case are λ
S= 12,I= 1
21,2 = 2,−1. Plugging
them into the transcendental equation for the S-wave scaling parameter s(0)i (Eq. (3.118)) one
obtains that only in the doublet channel s(0)1 = 1.00624 i is indeed purely imaginary. Hence,
the Efimov effect is present in the system. This result perfectly coincides with the work done inRefs. [80, 92].
4.1.2 KKK system
In the second detailed example we consider the system KKK. From experiment we know thatthere are two candidates for hadronic molecules in this system (cf. Tab. 4.1): a0(980) and f0(980).Both can be interpreted as a KK state, but with different quantum numbers [2]:
IG(JPC
)(a0) = 1−(0++)
IG(JPC
)(f0) = 0+(0++) . (4.3)
The kaons K themselves have I(JP ) = 1/2(0−). To answer the question whether or not anEfimov trimer exists in the system we will analyze a0–K scattering. Namely, following the rulesdescribed in the previous section we allocate the three kaons to the generic three particles P1,P2 and P3 as follows:
P1 = K ,
P2 = P3 = K . (4.4)
Because of P2 = P3 we have to consider just two dimers (cf. rules in the box on page 51): d12 andd′12. Since all three particles in the system have the same mass we consider the type 1 results,i.e. we have to find the matrix A1 and diagonalize it. From the allocation above we can directlyread off almost all parameters needed to determine the elements of A1. Indeed, Eq. (4.4) setsb1 = a2 = a3 = 1 and a1 = b2 = b3 = 0. Furthermore, we have
δP1P2 = δP1P3 = 0 ,
δP2P3 = 1 ,
δA1A2 = δA1A3 = δA2A3 = 1 , (4.5)
and thus
δ(ab(′))PiPj
≡ δPiPj , ∀ a < b ∈ 1, 2, 3 . (4.6)
75
It is sufficient to calculate v(′)12 and w
(′)12 since all elements with other indices are erased from A1.
With G-parity quantum numbers η12 = −1 and η′12 = +1 we find
v12 = 1×[
1√2
+ 1×(
1− 1√2
)](1× 1 + 0× 0)× (−1)1 + (1− 1)× 1× 1 = −1 , (4.7)
v′12 = +1 , (4.8)
w12 = w′12 = 0 , (4.9)
which is needed to determine f and f . Also the symmetry factors S12 = 1 and S123 = 2 as wellas S213 = 1 are a consequence of the particle allocation in Eq. (4.4). Obviously, there are nofermion minus signs (i.e. ζ123 = ζ213 = +1) since all three particles are bosons. The 2× 2 matrixA1 is thus given by
A1 =
x2c12
− x5√c12c′12
− x′2√c12c′12
x′5c′12
. (4.10)
The remaining four spin and isospin dependent parameters x2, x5, x′2 and x′5 can be calculatedusing one of the methods explained in appendix A. All methods have in common that they usenormalized projection operators and hence we know c12 = c′12 = 1. This holds independently ofthe scattering channel. However, the x parameters are of course different for different spin andisospin. While the spin channel is uniquely determined to be 0 the isospin one can be either1/2 or 3/2. In order to find the x parameters we use in this example their defining equationsin Eqs. (A.43 - A.52). Plugging in the right operator of Tab. A.2 to Tab. A.7 according to thespin and isospin quantum numbers of a0, f0, K and choose one of the two scattering channelsone has to calculate in the spin 0 & isospin 1/2 channel:
x2 =1
2
−1√3
(τg)λσi√2
(τmτ2)ρσ−i√
2(τ2τg)νρ
−1√3
(τm)νλ = −1
2,
x′2 =1
2δλσ
i√2
(τmτ2)ρσ−i√
2(τ2)νρ
−1√3
(τm)νλ =
√3
2,
x5 =1
2
−1√3
(τg)λσi√2
(τ2)ρσ−i√
2(τ2τg)νρδνλ =
√3
2,
x′5 =1
2δλσ
i√2
(τ2)ρσ−i√
2(τ2)νρδνλ =
1
2, (4.11)
and in the other possible channel with isospin 3/2 the task is to determine
x2 =1
4
1
3[(τgτ`)λσ + δg`δλσ]
i√2
(τmτ2)ρσ−i√
2(τ2τg)νρ
1
3[(τ`τm)νλ + δ`mδνλ] = 1 , (4.12)
since x′2 = x5 = x′5 = 0 because the isoscalar f0 cannot contribute to the second channel. Thenumbers above are obtained using the well-known identities for the Pauli matrices summarizedin appendix A.
76
The matrix A1 in the spin 0 & isospin 1/2 channel is given by
AS=0,I= 12
1 =
(−1
2−√
32
−√
32
12
), (4.13)
and its eigenvalues are λ1 = −1 and λ2 = 1 so that the transcendental equation for L = 0 inEq. (3.118) yields
s(0)1 = 2 (4.14)
s(0)2 = 0.413697 i . (4.15)
Since one solution is indeed purely imaginary we deduce that the Efimov effect is present, butits scaling factor exp(iπ/s
(0)2 ) = 1986.14 is rather large.
In the I = 3/2 channel one obtains just a single number, i.e. an 1× 1 matrix AS=0,I= 32
1 = 1 = λ1.
For S-wave scattering Eq. (3.118) has a purely imaginary solution s(0)1 = 0.413697 i so the Efimov
effect is again present with the same scaling factor of 1986.14 as for isospin 1/2 and we concludethat there must be in either channel a KKK trimer state in the system. As a remark, note thateven if the a0(980) is not a molecule and hence the f0(980) (whose molecular substructure ismore established [174]) is the only remaining dimer in the three kaon system, the scaling factorof 1986.14 is unchanged.
4.1.3 Summary of other established systems
After those two detailed examples we will now summarize the results for all physically promisingcombinations of molecules and single particles in Tab. 4.1. Namely, for all three particle systemswhere at least two of the three particles have a bound or virtual state. Depending on theconsidered particles we either apply the rules for type 1 or type 2 system. However, the in section 3derived methods do not provide a straightforward analysis for the following kind of systems: asystem of two particles with (approximately) equal mass and a third particle with different masswhere the equal mass ones have a resonant interaction (i.e. a bound or virtual state) cannotbe analyzed. Hence, we will use the mass difference parameters εij := (mi − mj)/(mi + mj)introduced in Eq. (3.84) to classify systems where the mass difference is still small enough sothat type 1 system equations are applicable. Therefore one has to compare in each relevantsystem the mass difference parameter and the effective range of the corresponding molecule. Thecontribution of the latter is already neglected as we have only considered leading order terms inthe effective range expansion. Unfortunately, in almost all two-body systems the effective rangeis not known (one exception is the NN system with the deuteron as bound state). Thus, wehave to assume that the corrections from setting εij ≈ 0 are indeed at most of the order of theeffective range corrections. This should be a justified assumption as it can be seen by comparisonwith Ref. [167] where the mass difference between nucleons and Λ particles (εN,Λ ≈ 0.1) is alsoneglected.In the same way as described above for the NNN and KKK system one finds for all othersystems the matrices which needs to be diagonalized in order to find their eigenvalues λi. Weplug these eigenvalues into the transcendental equation for the S- and P -wave scaling parameter
77
s(0,1)i given in Eqs. (3.118, 3.119) for type 1 or in Eqs. (3.131, 3.137) for type 2 systems. Note,
however, that it is shown in Refs. [80, 172, 173] that for symmetry reasons some boson/fermionconfigurations can only occur in even or odd partial wave channels, respectively. For instance ina three boson system the Efimov effect can only be present in the S-wave channel.In Tab. 4.2 we have summarized all systems where the Efimov effect occurs. One observes that– concerning fully bosonic systems – only in the KKK and in the scattering of charm mesonsoff the not so well-established KD1 molecule D∗s1(2700) the Efimov effect is present. Is themolecule in the latter scattering process a D∗s0(2317) or a Ds1(2460) instead one finds no Efi-mov trimer since the mass ratio ε12 = εK,D(∗) is not large enough to push the factor in front
of the trigonometric functions in the transcendental equation for s(0) above a critical value sothat s(0) becomes purely imaginary. Also in all (according to Tab. 4.1) possible combinations ofthree charm mesons or of three bottom mesons (both analyzed using the type 1 scheme sincethe mass difference is reasonable small) there is no Efimov effect. The conditions under whichsuch meson–meson molecules would be affected by Efimov physics will be discussed in the nextsection.Next, we focus on systems consisting of fermions only. Besides the already known results con-cerning the NNN [80, 92] and NNΛ system [167] one finds evidence for Efimov trimers in thescattering of a nucleon off a H-dibaryon (ΛΛ) which was suggested by Jaffe in Ref. [175]. Afurther extension with more dibaryon states like NΣ, NΞ, ΛΣ, ΛΞ, ΣΣ, ΣΞ, ΞΞ or even with Ωcombinations could be possible, but there is up to now no experimental evidence for such dimers.However, we will discus some issues concerning them in the next section.As a third alternative there could be molecules consisting of a baryon and a meson. An ex-ample for such a two-body system is the Λ(1405) lying just below the KN threshold so thatits interpretation as a virtual kaon–nucleon state is justified (see also Refs. [176, 177]). The twocorresponding three particle states are thus NKK and KNN . The former fits into the type 2 sys-tem scheme, but is not affected by Efimov physics. Unfortunately, the latter cannot be analyzedbecause the two equal mass particles (the nucleons) can form a bound (3S1 partial wave) and avirtual (1S0 partial wave) state. However, if one chooses one specific isospin channel, namely theone corresponding to the system K−pp, the type 2 method is applicable to KNN since the twoprotons have no dimer state. This gives us the opportunity to check whether the theoreticallypredicted [178–180], but not yet experimentally confirmed [181,182] K−pp three particle boundstate can be interpreted as Efimov trimer in the scattering of a proton off a Λ(1405). Indeed,we found in the KNN isospin 1/2 channel with I3 = +1/2 fixed that there is an Efimov effectwhich gives rise to a K−pp trimer. Its existence would thus support the interpretation of theΛ(1405) as hadronic molecule.
78
system channel scaling parameter Im(s(0))
scaling factor exp(
i πs(0)
)
KKKS = 0 & I = 1
20.413697 1986.14
S = 0 & I = 32
0.413697 1986.14
KD1D (?) S = 1 & I = 12
0.231624 777104.0
KD1D∗ (?)
S = 0 & I = 12
0.231624 777104.0
S = 1 & I = 12
0.231624 777104.0
S = 2 & I = 12
0.231624 777104.0
KD1D1 (?)
S = 0 & I = 12
0.231624 777104.0
S = 1 & I = 12
0 S = 2 & I = 1
20.231624 777104.0
NNN
S = 12
& I = 12
1.00624 22.69
S = 12
& I = 32
0 S = 3
2& I = 1
20
NNΛ
S = 12
& I = 0 1.00624 22.69
S = 12
& I = 1 1.00624 22.69
S = 32
& I = 0 1.00624 22.69
S = 32
& I = 1 0
NΛΛS = 1
2& I = 1
21.00624 22.69
S = 32
& I = 12
0
K−ppS = 0 & I = 1
20.605355 179.41
S = 1 & I = 12
0.605355 179.41
Table 4.2: Overview of the considered three particle systems according to Tab. 4.1 in whichthe Efimov effect is present. Shown is the imaginary part of the (in this case purely imaginary)S-wave scaling parameter s(0) and the corresponding scaling factor exp
(i πs(0)
). Note, that there
is no P -wave Efimov trimer because of symmetry reason such a state can only be present forspecific boson / fermion configuration as it is shown in Refs. [80, 172, 173]. The question markindicates that the D∗s1 = KD1 molecule interpretation is very hypothetical.
79
4.2 Hypothetical systems
After our analysis of a large number of more or less established molecules scattering off a thirdparticle we now consider some hypothetical two-body bound states in order to identify the spinand isospin quantum numbers which would lead to Efimov trimers in corresponding scatteringprocesses. For all these systems we assume that one indeed can treat them in EFT(/π). Namely,we assume that their binding momenta are at most of the order of the pion mass so that one canobtain at least some first estimates (see also section 4.3). We will focus on three different systems:firstly, one can assume that besides the known DD (BB) charm and bottom mesons also DD(BB) bound states exist (a possible explanation for the fact that they are not yet discoveredat one of the B-factories would be that one needs at least four charm or bottom mesons sincethey are produced in anti-particle–particle pairs). Secondly, one can consider the known charmand bottom molecules, but change their quantum numbers to other values. Thirdly, one canconsider the large number of hypothetical dibaryon bound states. However, all these two-bodysystems have in common that – choosing an appropriate third particle scattering off the molecule– the corresponding three-body systems consists of identical particles or at least of multiplet–anti-multiplet combinations. Therefore the number of parameters in our type 1 system analysesreduces and additionally the remaining ones are more restricted.
4.2.1 Hypothetical charm and bottom meson systems
Assuming that there is a shallow dimer in the system of two charm or bottom mesons, respectively,one can consider the systems DDD, D∗D∗D∗ or D1D1D1 in the charm sector and BBB, B∗B∗B∗
or B1B1B1 in the bottom meson sector. In each system one deals with identical particles.According to the rules in the box on page 51 one thus has to erase the amplitudes T13 and T23
from type 1 system equations. The matrix A1 defined in Eqs. (3.99 - 3.104) is thus reduced to a2× 2 matrix Aid
1 whose elements are:
(Aid
1
)11
=
(1− δP1P2
2
)1
S12 c12
[x2 δ
(12)P1P3
f(3)(12)(12) S123 + x2 δ
(12)P2P3
f(3)(12)(12) S213
],
(Aid
1
)21
=
(1− δP1P2
2
)1
S12
√c′12 c12
[x′2 δ
(12)P1P3
f(3)(12)(12′) S123 + x′2 δ
(12)P2P3
f(3)(12)(12′) S213
],
(Aid
1
)12
=
(1− δP1P2
2
)1
S12
√c12 c′12
[x5 δ
(12′)P1P3
f(3)(12′)(12) S123 + x5 δ
(12′)P2P3
f(3)(12′)(12) S213
],
(Aid
1
)22
=
(1− δP1P2
2
)1
S12 c′12
[x′5 δ
(12′)P1P3
f(3)(12′)(12′) S123 + x′5 δ
(12′)P2P3
f(3)(12′)(12′) S213
]. (4.16)
From this result it is straightforward to deduce the entries of Aid1 if there are more than two
different spin/isospin configurations, that is, if there are dimers d′′12, d′′′12, etc. with other quantumnumbers. But firstly, we notice that all (normal and modified) Kronecker-deltas yield 1 for three
identical particles P1 = P2 = P3. Furthermore, f(.)(..)(..) = f
(.)(..)(..) = 1 independently of the
arguments since we have ai = 1 for i = 1, 2, 3. Also the symmetry factor S12 = 2 is already fixed.
80
Only the second symmetry factor S123 = S213 = ±4 is not unique; it is positive in the case thatthe three identical particles are bosons, but negative for fermions. One can thus write for anarbitrary number of dimer states:
(Aid
1
)ab
= ± 1√c
[(a−1)′]12 c
[(b−1)′]12
[x
[(a−1)′]2b+(b−1) + x
[(a−1)′]2b+(b−1)
], (4.17)
where the plus (minus) sign is for bosons (fermions) and the notation [(a− 1)′] has to be under-stood in the manner of [3 ′] = ′′′ and so on. In order to get rid of the remaining spin/isospinparameters we use the 6-J symbol notation explained in appendix A.3. From Eqs. (A.111 -
A.146) we deduce with δP1P2 = 1 that the parameters x[(a−1)′]2b+(b−1) and x
[(a−1)′]2b+(b−1) are equal and given
by
x[(a−1)′]2b+(b−1) = x
[(a−1)′]2b+(b−1) = (−1)2S
√(2J
[(a−1)′]12 + 1
)(2J
[(b−1)′]12 + 1
)j1 j1 J[(a−1)′]12
j1 S J[(b−1)′]12
× (−1)2I
√(2I
[(a−1)′]12 + 1
)(2I
[(b−1)′]12 + 1
)i1 i1 I[(a−1)′]12
i1 I I[(b−1)′]12
, (4.18)
with j1 (i1) being the spin (isospin) of the identical particles P1 = P2 = P3 and S (I) is theconsidered spin (isospin) channel. Since we have derived the equation above using normalizedspin and isospin projectors we know that the factors c12 are equal to 1 independently of thenumber of primes. Hence, Eq. (4.17) can be written as
(Aid
1
)ab
= ± 2(−1)2(S+I)
√(2J
[(a−1)′]12 + 1
)(2J
[(b−1)′]12 + 1
)√(2I
[(a−1)′]12 + 1
)(2I
[(b−1)′]12 + 1
)
×j1 j1 J
[(a−1)′]12
j1 S J[(b−1)′]12
i1 i1 I
[(a−1)′]12
i1 I I[(b−1)′]12
. (4.19)
Consequently, the existence of the Efimov effect is completely determined by the spin and isospinstructure of the particles and the dimers. The question is now: what do we know about thesedegrees of freedom? Firstly, we know from Tab. 4.1 that all charm and bottom mesons are isospin1/2 particles and secondly, the spin is either 0 for D, B or 1 for D∗, D1, B∗, B1. Thus, oneconcludes that on the one hand both pseudoscalar charm (DDD) and bottom (BBB) mesonsystems behave exactly the same and on the other hand that the remaining vector (D∗D∗D∗,B∗B∗B∗) and axialvector (D1D1D1, B1B1B1) meson systems are equivalent in the sense of Efi-mov physics. To find the spin and isospin configurations leading to an Efimov effect it is thusnecessary to identify all possible two-body quantum numbers J12 and I12 of the dimers. Fromthe constituent particles we conclude that for all systems i1 ⊗ i1 = 1⊕ 3, i.e. that there are twomolecule states: one with I12 = 1 and one with I ′12 = 0. Concerning spin one has to distinguishbetween D or B systems and D∗, B∗, D1 or B1 systems. All in all one finds the combinationsshown in Tab. 4.3 where also the matrix Aid
1 is given. Calculating the eigenvalues for both sys-tems and for all channels one concludes that for all spin and isospin configurations except forthe S = 0 & I = 3/2 channel of the (axial-)vector system there is always at least one eigenvalue
81
λid = 2. Plugging this into the transcendental equation for the S-wave scaling parameter s(0),Eq. (3.118),
1 =4λid√
3
1
s(0)
sin(π6s(0))
cos(π2s(0)) ,
one finds that s(0) = 1.00624 i is a solution. Hence, a system of three identical charm or bottommesons behaves in all channels – except for the mentioned S = 0 & I = 3/2 channel of the(axial-)vector system – like a three identical boson system with spin- and isospinless particles.This statement is only true if all theoretically possible two-body spin and isospin configurationsindeed exist in nature. Considering for example the DDD or BBB pseudoscalar systems oneobserves that in the case of only a I12(J12) = 0(0) dimer the system has a scaling factor ofs(0) = 0.413697 i. In the case of only an I12(J12) = 1(0) dimer one finds s(0) = 1.00624 i in theS = 0, I = 3/2 channel, but there is no Efimov effect in the other allowed S = 0, I = 1/2channel. Since in the (axial-)vector system of charm and bottom mesons the number of present/ missing two-body configurations is large we will not discuss all of them for every scatteringchannel. Instead, we give some qualitative results concerning the existence of the Efimov effect:
Qualitative results for hypothetical identical charm and bottom meson systems
DDD and BBB systems:
• One or two dimers: in a system of three identical pseudoscalar charm or bottom mesonsthe Efimov effect is always present in at least one scattering channel independently ofthe number and of the quantum numbers of the dimer or the two dimers, respectively.
D∗D∗D∗, D1D1D1, B∗B∗B∗ and B1B1B1 systems:
• Exactly one dimer: in a three identical spin 1 charm or bottom meson system theEfimov effect does not occur if this dimer has the quantum numbers I12(J12) = 0(0), 1(0)or 0(1). For the other allowed configurations (1(1), 0(2) and 1(2)) one always finds atleast one scattering channel where Efimov physics are relevant.
• Exactly two dimers: in such a system the Efimov effect does not occur if the twodimers have the quantum numbers I12(J12) = 0(0) and I ′12(J ′12) = 1(0) or vice versa. Allother combinations lead in at least one scattering channel to an Efimov trimer in thesystem.
• More than two dimers: for every configuration of dimers and quantum numbers therewill be an Efimov trimer in at least one scattering channel.
The conclusion is thus that – if these kind of molecules indeed exist in nature – it is very likelythat the Efimov effect is an important property of such a system of three identical charm orbottom meson systems.
82
j1 dimer QN channel QN matrix Aid1
j1 = 0J12 = 0 & I12 = 0
J ′12 = 0 & I ′12 = 1
S = 0 & I = 12
1 −
√3
−√
3 −1
S = 0 & I = 32
2
j1 = 1
J12 = 0 & I12 = 0
J ′12 = 0 & I ′12 = 1
J ′′12 = 1 & I ′′12 = 0
J ′′′12 = 1 & I ′′′12 = 1
J ′′′′12 = 2 & I ′′′′12 = 0
J ′′′′′12 = 2 & I ′′′′′12 = 1
S = 0 & I = 12
−1
√3
√3 1
S = 0 & I = 32
-2
S = 1 & I = 12
13− 1√
3− 1√
31
√5
3−√
5√3
− 1√3−1
31 1√
3−√
5√3−√
53
− 1√3
1 12
−√
32
√5√12
−√
52
1 1√3−√
32
−12−√
52−√
5√12√
53−√
5√3
√5√12
−√
52
16
− 1√12
−√
5√3−√
53−√
52−√
5√12− 1√
12−1
6
S = 1 & I = 32
23− 2√
32√
53
− 2√3
1√
5√3
2√
53
√5√3
13
S = 2 & I = 12
12−√
32−√
32
32
−√
32−1
232
√3
2
−√
32
32
−12
√3
2
32
√3
2
√3
212
S = 2 & I = 32
1 −
√3
−√
3 −1
S = 3 & I = 12
1 −
√3
−√
3 −1
S = 3 & I = 32
2
Table 4.3: All possible spin and isospin configurations (QN: quantum numbers) for the pseu-doscalar (j1 = 0) meson systems DDD, BBB and for the three spin j1 = 1 charm and bottommeson systems D∗D∗D∗, D1D1D1, B∗B∗B∗, B1B1B1. The isospin of the identical particlesP1 = P2 = P3 is always i1 = 1/2 and not explicitly given.
83
4.2.2 More dimers in existing charm and bottom meson systems
After the discussion of the completely hypothetical three identical meson systems in the previoussubsection we will now consider the experimentally confirmed states in Tab. 4.1. We assume thatthere are more bound states in the underlying two-body molecule systems with different spin andisospin configurations. This is motivated by the fact that the processes in which the molecules areproduced in experiments may favor some or even forbid other configurations because of symmetry.However, this argument makes sense only for (axial-)vector charm and bottom mesons, i.e. D∗,D1, B∗ and B1 because for the molecules with one pseudoscalar meson instead we have alreadychecked in section 4.1.3 that this does not lead to an Efimov effect. This was explicitly donefor the charm molecules X(3872) and Zc(3900) and thus we know that – a not yet discovered –iso-singlet partner of the Zb(10610) would not change the situation in the bottom sector. Thisis clear because except for their masses all relevant quantum numbers for Efimov physics of Dand B as well as of D∗ and B∗ are equal. Since there are no D1D1 or B1B1 dimers known so farwe restrict ourselves to the vector charm and bottom mesons which behave in the sense of theEfimov effect completely the same. Hence, it will be sufficient to analyze only one system; theresult will be valid for both D∗D∗ and B∗B∗.Consequently, we consider the three particle system P1P2P3 with P1 = A1, P2 = P3 = A1, whereA1 is an isospin i1 = 1/2 and spin j1 = 1 field (corresponding to both D∗ and B∗). A two-bodybound state between A1 and A1 can thus have the following spin and isospin quantum numbers:
J12 = 0 & I12 = 0, J ′12 = 0 & I ′12 = 1, J ′′12 = 1 & I ′′12 = 0,
J ′′′12 = 1 & I ′′′12 = 1, J ′′′′12 = 2 & I ′′′′12 = 0, J ′′′′′12 = 2 & I ′′′′′12 = 1 .
Although these are the same combinations as in the previous section on three identical mesonsone has to keep in mind that the situation is different. On the one hand particles P1 and P2 arenot identical and on the other hand not all particles have a large scattering length (we do notconsider a hypothetical A1A1 dimer in contrast to what was done above). Indeed, our particleallocation tells us that b1 = a2 = a3 = 1 and η12 = ±1 for all numbers of primes. Therefore onefinds v12 = η12 and f
(3)
(12)(′)(12)[′] = f(3)
(12)(′)(12)[′] = a3v(′)12v
[′]12 = η
(′)12η
[′]12 which also holds for all numbers
and combinations of primes. Concerning the modified Kronecker-deltas one gets δ(12)P1P3
= 0 andδP2P3 = 1 and similarly for primed symbols. Finally, the relevant symmetry factors are S12 = 1,S123 = +2 and S213 = +1 as we deal with bosons. All together and with the ”multiple prime“notation from the previous section 4.2.1 this leads to a matrix AAAA1 whose elements are givenby
(AAAA1
)ab
= x[(a−1)′]2b+(b−1) η
[(a−1)′]12 η
[(b−1)′]12
= η[(a−1)′]12 η
[(b−1)′]12 (−1)2(S+I)(−1)J
[(a−1)′]12 +J
[(b−1)′]12 +I
[(a−1)′]12 +I
[(b−1)′]12 −4−2
×√(
2J[(a−1)′]12 + 1
)(2J
[(b−1)′]12 + 1
)√(2I
[(a−1)′]12 + 1
)(2I
[(b−1)′]12 + 1
)
×j1 j1 J
[(a−1)′]12
j1 S J[(b−1)′]12
i1 i1 I
[(a−1)′]12
i1 I I[(b−1)′]12
. (4.20)
84
Here, Eqs. (A.111 - A.146) were used to replace the spin and isospin dependent x parameters by6-J symbols. Besides a minus sign from the additional phase factor and/or the G-parity quantumnumbers in front, the matrix elements are half as large as those in the identical charm or bottommeson system (cf. Eq. (4.19)). Hence, one could make a similar table like Tab. 4.3 in order todeduce that although some elements indeed change their sign the eigenvalues does not. However,they are divided by two, i.e. λid = λAAA/2. In Refs. [80,166] it is shown that
1 =4λ√
3
1
s(0)
sin(π6s(0))
cos(π2s(0))
has a purely imaginary solution (i.e. the Efimov effect occurs) if the eigenvalue λ is larger
than a critical value λC = 3√
32π≈ 0.826993. Therefore the extra factor of 1/2 compared to the
eigenvalues in D∗D∗D∗ or B∗B∗B∗ systems is an important difference. Indeed, one observes thatthe qualitative results concerning the existence of the Efimov effect are changed in comparisonto the previously discussed case:
Qualitative results for extended D∗D∗ or B∗B∗ hadronic molecules
• Exactly one dimer: in a D∗D∗D∗ or B∗B∗B∗ system the only dimer quantum numbersfor which the Efimov effect is present (exclusively in the S = 3, I = 3/2 channel) areI12(J12) = 1(2). For all other configurations (0(0), 1(0), 0(1), 1(1) and 0(2)) Efimovphysics are not relevant.
• Exactly two dimers: an Efimov trimer can be found if the two dimers have the quantumnumbers I12(J12) = 1(1) and I ′12(J ′12) = 0(1), 1(0) or vice versa. In all other cases thereis no Efimov effect.
• Exactly three dimers: in such a system one finds an Efimov trimer in at least onescattering channel for the dimers with all possible quantum numbers except for I12(J12) =0(0), I ′12(J ′12) = 0(1) and I ′′12(J ′′12) = 0(2) or any permutation of these regarding theprimes.
• More than three dimers: for all dimer configurations and for all quantum numbersthe Efimov effect will be present in at least one scattering channel.
As it is written in Tab. 4.1 we know that both Z ′b = B∗B∗ and its charm analog Z1 = D∗D∗ havethe quantum numbers I(J) = 1(1). Therefore one concludes that as soon as an additional D∗D∗
or B∗B∗ state with either I ′(J ′) = 0(1) or 1(0) or 1(2) is discovered, Efimov physics should betaken into account in Z1–D∗ and Z ′b–B
∗ scattering.
85
4.2.3 Hypothetical dibaryons
Now we focus on fermionic systems: besides the H-dibaryon called ΛΛ bound state which waspredicted by Jaffe in Ref. [175] there are much more candidates for dibaryon states like ΣΣ orΞΞ, but also in the charmed or bottom baryons sector there could be states like ΛcΛc. See forexample Refs. [183,184] for a theoretical discussion of some of the candidates and Refs. [185,186]for lattice calculations regarding dibaryons with Ω’s. Thus, it may be worth the effort to analyzethree identical fermion systems like ΣΣΣ, ΞΞΞ and so on. One has to deal with a system ofidentical particles which can be described using the type 1 method. Moreover, the correspondingmatrix A1 is already known from section 4.2.1 where we have derived it for three identicalparticles independently of their species:
(Aid
1
)ab
= ± 2(−1)2(S+I)
√(2J
[(a−1)′]12 + 1
)(2J
[(b−1)′]12 + 1
)√(2I
[(a−1)′]12 + 1
)(2I
[(b−1)′]12 + 1
)
×j1 j1 J
[(a−1)′]12
j1 S J[(b−1)′]12
i1 i1 I
[(a−1)′]12
i1 I I[(b−1)′]12
, (4.21)
where the plus sign is valid for bosons and the minus sign for fermions. All dibaryon–baryonscattering processes can thus be described using the relation above with a minus sign in front.However, the spin and isospin quantum numbers are less restricted than in the three charm orbottom meson case in section 4.2.1. Independently of additional charm or bottom quarks (seefor example Ref. [2]) Λ(c,b) ground state baryons have the quantum numbers I(J) = 0(1/2),Σ(c,b) ground state baryons have I(J) = 1(1/2) and Ξ(c,b) ground states have spin and isospinI(J) = 1/2(1/2). Only Ω baryons have different quantum numbers if one strange quark isreplaced by a charm or bottom one. Namely, for the ground state one has I(J) (Ω) = 0(3/2),but I(J) (Ωc,b) = 0(1/2). Hence, we notice that the Ωc,b Ωc,b Ωc,b system is in the sense of Efimovphysics equivalent to the Λ(c,b) Λ(c,b) Λ(c,b) system. Additionally, we deduce that the spin andisospin quantum numbers of the three Σ(c,b) system are interchanged compared to the D∗D∗D∗,B∗B∗B∗, etc. systems. Consequently, the eigenvalues in the latter fermionic system have thesame modulus, but the opposite sign as in the bosonic system. In the following we summarizethe qualitative results in the same way as it was done before:
Qualitative results for hypothetical dibaryons
Λ(c,b)Λ(c,b)Λ(c,b) and Ωc,bΩc,bΩc,b systems:
• Exactly one dimer: if spin and isospin of the dimer are I12(J12) = 0(0) one finds noEfimov trimer while for I12(J12) = 0(1) the Efimov effect occurs in the S = 1/2, I = 0channel.
• Exactly two dimers: if both allowed dimer states with quantum numbers I12(J12) =0(0) and I ′12(J ′12) = 0(1) are present in the system, Efimov physics are relevant in theS = 1/2, I = 0 scattering channel (cf. ΛΛΛ system in Tab. 4.2).
86
Σ(c,b)Σ(c,b)Σ(c,b) systems:
• Exactly one dimer: for the dimer quantum numbers I12(J12) = 1(0), 1(1) and 2(1)there will be at least one scattering channel with an Efimov trimer while for dimers withI12(J12) = 0(0), 0(1) and 2(0) Efimov physics are not relevant.
• Exactly two dimers: except for the two dimers with quantum numbers I12(J12) = 0(0)and I ′12(J ′12) = 0(1), 2(0) or the dimers with I12(J12) = 0(1) and I ′12(J ′12) = 2(0), onefinds for all other combinations at least one channel where the Efimov effect occurs.
• More than two dimers: for every possible number of dimers larger than two there isalways at least one scattering channel with an Efimov trimer.
Ξ(c,b)Ξ(c,b)Ξ(c,b) systems:
• Exactly one dimer: only for a dimer with quantum numbers I12(J12) = 1(1) the Efimoveffect is present in the S = 1/2, I = 3/2 and in the S = 3/2, I = 1/2 channel. For thethree other possible dimers one finds no Efimov effect.
• More than one dimer: for an arbitrary number of dimers larger than two one findsindependently of the quantum numbers at least one scattering channel with an Efimovtrimer.
ΩΩΩ systems:
• Exactly one dimer: for a dimer with spin and isospin I12(J12) = 0(0) one finds noEfimov trimer, but for all other dimers with quantum numbers I12(J12) = 0(1), 0(2) or0(3) the Efimov effect occurs in the S = 1/2, S = 3/2 or S = 7/2 scattering channel(with I = 0 being unique).
• More than one dimer: if there is more than one dimer in the system Efimov physics arerelevant in at least one scattering channel independently of the dimer quantum numbers.
In conclusion, one deduces that Efimov physics are an important phenomenon in dibaryon–baryon scattering and if dibaryon molecules are found in experiments it would be promising tosearch for three particle bound states in order to clarify their substructure.
4.3 Summary of the results
In this last part of the current section we will summarize and give some remarks concerningthe results above. First of all we must once again emphasize that the substructure of many ofthe considered particles is not clear. Hence, their interpretation as hadronic molecules is just ahypotheses. However, if this interpretation is correct their treatment in a non-relativistic pionlesseffective field theory is justified as long as their binding momenta are smaller than the pion massor at least of that order. Calculating the binding momentum γij = sig(Bij)
√2µij|Bij| using
the binding energy determined with the relation Bij = (mi + mj) −Mdij one observes that the
87
mentioned condition is on the one hand clearly fulfilled for a number of particles in Tab. 4.1 likethe X(3872) with binding momentum γX ∼ 15 MeV. On the other hand there are also particleslike the Zc(3900) whose binding momentum |γZc | ∼ 211 MeV is even larger than the pion mass.However, especially due to the often large uncertainties regarding the masses of the moleculecandidates it should be justified to use EFT(/π) also for the ”problematic“ particles in order toobtain at least some first insights to the respective molecule–particle scattering. In the same waywe assumed that the binding momenta of the considered hypothetical states are also at most ofthe order of mπ so that their treatment in EFT(/π) is possible. Of course, future experimentscould prove this assumption wrong. As a remark, note that one could add pions to the theory toget a more accurate description of the physical system like it was done for the X(3872) using theso-called XEFT in Refs. [187,188]. Regarding the Efimov effect we found that only a rather smallnumber of molecule–particle scattering processes is affected by an intermediate Efimov trimer(see Tab. 4.2). However, experimental setups possibly restrict the allowed quantum numbers of apossible molecular state. Thus, there might be a reasonable chance that more states with differentquantum numbers exist in nature. Depending on the exact spin and isospin of these additionalmolecules we have shown that the Efimov effect might become important (see box ”Qualitativeresults for extended D∗D∗D∗ or B∗B∗B∗ hadronic molecules“ on page 85). Moreover, there mightexist charm or bottom meson molecules with identical constituents (DDD, B∗B∗B∗, etc.) forwhich we have checked that their scattering off a third (identical) particle is most likely affectedby Efimov physics. In fact, only if there is just one dimer with I12(J12) = 0(0), 1(0), 0(1) thiswill not be the case (see box ”Qualitative results for hypothetical identical charm and bottommeson systems“ on page 82). From this point of view it would be very interesting to search formore molecule states of the ”anti-charm/bottom meson – charm/bottom meson“ type and for thecompletely new ”charm/bottom meson – charm/bottom meson“ states. While for the first typethe existing e+e− collider B-factories are not favorable due to symmetry reasons which forbid orsuppress the production of other quantum numbers, they are in principle suitable for the lattertype. However, one would need rather high energies since the charm and bottom mesons areproduced in particle–anti-particle pairs. Consequently, one would need four bottom mesons intotal in order to observe e.g. a BB molecule. Furthermore, we have also found that dibaryonsystems are – with just a few exceptional quantum numbers – affected by the Efimov effect (seebox ”qualitative results for hypothetical dibaryons“ on page 86). Hence, more experimental ef-fort to discover such states could be worthwhile.A common feature in all cases is that experimentally observing a molecule–particle scatteringprocess which is affected by an Efimov trimer would strongly support the molecule interpreta-tion of the accordant particle. Moreover, discovering even a second Efimov trimer with bindingenergy (exp(iπ/s))2 times larger could be seen as almost a proof of the molecular nature of thecorresponding particle. However, it is not clear whether there are observable second trimer statesat all since the Efimov spectrum is cut off at a certain point as explained in section 2.1.
An important remark concerning this work is that – although we exclusively have focused onthem – it is not restricted to hadronic molecules. The derivation in section 3 was completelygeneral and hence one can apply the presented methods to every particle scattering off a S-wavedimer as long as either for an arbitrary number of dimer states all masses are (approximately)equal (type 1 system) or for m1 6= m2 = m3 if there is no d23 dimer (type 2 system). Not
88
analyzable with our method are systems with three different particle masses and systems whereexactly two of the three particles have equal mass (and the third an unequal one), but where theequal mass particles also have a dimer state. However, apart from these subtleties the derivedmatrices A1 (type 1) and A2 (type 2) are also valid in the physics of cold atoms or halo nuclei.
As a final remark, we want to emphasize that in all three-body systems of equal mass whichwe have discussed, the maximal eigenvalue was λ = 2 corresponding to a scaling parameter ofs = 1.00624 i. Thus, it seems to be the case that there is some kind of minimal scaling factorexp(π/1.00624)2 ≈ 515.03 in Efimov physics. A further analysis in order to proof or falsify thisconjecture and if it is true to check its possible extension to the unequal mass system should beworthwhile.
89
Chapter 5
Elastic molecule–particle scatteringobservables
As it was shown in the previous chapter the up to now known charm and bottom meson moleculesare not affected by Efimov physics if they are scattered off a third charm or bottom meson. How-ever, the absence of Efimov trimers makes the respective scattering processes elastic. Thereforeone can extract from the given S-wave three-body scattering amplitude the scattering lengthand the phase shift as three-body observables.At this point it is useful to recapitulate the cutoff independence of these observables: in chap-ter 2.1 it was explained that the existence of a three-body bound state is in some sense indepen-dent of the exact value of the three-body coupling constant H(Λ) which thus can be set to zero(according to the argument that there always exists a cutoff Λ0 so that H(Λ0) = 0). However,some of the observables are – directly or indirectly – affected by a three-body bound state andtherefore via H cutoff dependent. This means that if a three particle bound state is found, thethree-body binding energy B3 and scattering length a3 are functions of Λ. This dependence canonly be removed using an experimentally measured value of either B3 or a3 to fix H(Λ) whichthen allows a prediction of the respective other. In contrast, if there is no three-body boundstate then one concludes that there is no three-body force at all. For energies |E| < Λ the re-maining scattering observables are thus not cutoff dependent. In particular, this fact allows us totreat molecule–particle scattering at negative energies below the dimer threshold as a two-bodyscattering problem (see section 5.2.1). With an existing three-body bound state the situationwould change. According to the Efimov effect such a state would lie below the two-body bindingenergy. Thus, the system of a molecule and a particle could form such an Efimov trimer andhence their scattering would not be elastic anymore.As a selected example we will discuss the elastic scattering of B and B∗ meson off the twomolecules Zb(10610) and Z ′b(10650) in more detail. All other scattering processes without Efi-mov trimer, i.e. all elastic processes, could be analyzed in the same way.
90
5.1 Elastic S-wave Z(′)b –B(∗) scattering
In total there are four different scattering processes where the bottom mesons B and B∗ areinvolved: following Tab. 4.1 one has to consider Zb–B, Zb–B
∗, Z ′b–B and Z ′b–B∗ scattering.
Consequently, one has to deal with the three particle systems BB∗B, BB∗B∗, B∗B∗B andB∗B∗B∗ which will be discussed below. Regarding the notation we will use m and m∗ for themasses of B, B and B∗, B∗ mesons, respectively. Furthermore, µ (γ) is used for the reducedmass (the binding momentum) of the molecule Zb(10610) and µ′ (γ′) for the reduced mass (thebinding momentum) of Z ′b(10650).
5.1.1 Zb–B scattering amplitude
According to the method derived in chapter 3 we allocate the particles Pi as follows: P1 = P3 = Band P2 = B∗. Hence, we deduce a1 = b2 = a3 = 1 as well as
δP1P2 = δP2P3 = δA1A2 = δA2A3 = 0 ,
δP1P3 = δA1A3 = 1 . (5.1)
Furthermore (cf. Tab. 4.1), the G-parity quantum number is η12 = +1. Firstly, there is noresonant interaction with large scattering length between the two B mesons and secondly, thetwo dimers d12 and d23 are identical. Thus, one has to – according to the rules in the box onpage 51 – erase the latter from the equations. Therefore it remains just one dimer in the system(cf. Eq. (3.13)),
d12 =1√2
(B∗B +B∗B
), (5.2)
which represents the molecule Zb(10610) with IG(JP ) = 1+(1+). The only remaining amplitude
is thus T(L)12 and since we are interested in the scattering process one has to consider the amplitude
in Eq. (F.4) where the asymptotic momentum limit is not yet applied. For S-wave scatteringone finds
T(0)12 (E, k, p) =
2 π γ12
µ212 S12 c12
×[x1 δ
(12)P1P3
f(3)(12)(12) S123
m2
kpQ211
0 (k, p;E) + x1 δ(12)P2P3
f(3)(12)(12) S213
m1
kpQ122
0 (k, p;E)
]
+1
π
1
µ12 S12 c12
∫ ∞
0
dqq2 T
(0)12 (E, k, q)
−γ12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
×[x2 δ
(12)P1P3
f(3)(12)(12) S123
m2
qpQ211
0 (q, p;E) + x2 δ(12)P2P3
f(3)(12)(12) S213
m1
qpQ122
0 (q, p;E)
].
(5.3)
91
Using the summary on pages 56 and 57 all other parameters in T(0)12 are obtained to be given as
δ(12)P1P3
= 1 , δ(12)P2P3
= 0 ,
v12 = 1√2, w12 = 1√
2,
f(3)(12)(12) = a3w12w12 = 1
2, f
(3)(12)(12) = a3v12v12 = 1
2,
S12 = 1 , c12 = 1 ,
S123 = 1 , S213 = 2 ,
spin 1 & isospin 1/2 channel: x1 = x2 = −12, x1 = x2 = +1
2,
spin 1 & isospin 3/2 channel: x1 = x2 = +1 , x1 = x2 = −1 ,
where we have used in the last two lines Eqs. (A.111, A.129) together with the fact that B is apseudoscalar and B∗ is vector isospin doublet so that the allowed channels in Zb–B scatteringare spin 1 and either isospin 1/2 or isospin 3/2. Plugging the results into Eq. (5.3) yields
T(0)12 (E, k, p) =
−1
2
1
πγ12 m2
µ212
1
kpQ211
0 (k, p;E)
+
−1
2
1
1
2π
m2
µ12
∫ ∞
0
dqq2 T
(0)12 (E, k, q) 1
qpQ211
0 (q, p;E)
−γ12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
, (5.4)
where the upper entry corresponds to the isospin 1/2 and the lower one to the isospin 3/2channel. One can now reinsert the short-hand notation for the Legendre function of the secondkind defined in Eq. (3.73) and additionally use Eq. (D.10) which relates it to the logarithm (forthe behavior in the limit ε→ 0 see Eq. (D.15) which was derived in appendix D) in order to findthe representation of the amplitude below:
T(0)12 (E, k, p) =
−1
2
1
πγ m∗µ2
1
2kpln
[k2
2µ+ p2
2µ− E + kp
m∗− iε
k2
2µ+ p2
2µ− E − kp
m∗− iε
]
+
−1
2
1
1
2π
m∗µ
∫ ∞
0
dqq2 T
(0)12 (E, k, q)
−γ +
√−2µ
(E − q2
2m− q2
2(m+m∗)
)− iε
× 1
2pqln
[q2
2µ+ p2
2µ− E + pq
m∗− iε
q2
2µ+ p2
2µ− E − pq
m∗− iε
]. (5.5)
Note, that the upper (lower) component still represents isospin 1/2 (isospin 3/2) as before andthat we have replaced m2 by m∗, m1,3 by m, µ12 by µ being the reduced mass of the Zb(10610)and γ12 by γ being its binding momentum.
5.1.2 Zb–B∗ scattering amplitude
The second process we want to analyze is S-wave Zb–B∗ scattering. Hence, the scalar iso-doublet
B is replaced by the vector B∗. The corresponding three particle system BB∗B∗ is allocated to
92
the generic particles Pi as follows: P1 = B, P2 = B∗ and P3 = B∗. Thus, a1 = b2 = a3 = 1 asbefore, but in contrast to the Zb–B case it holds
δP1P2 = δP1P3 = δP2P3 = δA1A2 = δA1A3 = 0 ,
δA2A3 = 1 . (5.6)
While both two-body systems BB∗ and B∗B∗ are G-parity eigenstates (leading to δη12|1| =δη23|1| = 1), the combination BB∗ is not and hence δη13|1| = 0. According to Eq. (3.13) there arethree different dimers in the system:
d12 =1√2
(B∗B + η12B
∗B), (5.7)
d13 = BB∗ , (5.8)
d23 = η23B∗B∗ . (5.9)
From Tab. 4.1 we deduce that only d12 = Zb(10610) and d23 = Z ′b(10650) are present in nature.Thus, one can on the one hand insert the quantum numbers IG(JP ) = 1+(1+) for both statesand on the other hand erase all d13 contributions. Since there are no states with other quantumnumbers than 1+(1+) one has to erase all primed amplitudes, too (see page 51). The coupled
scattering amplitudes T(L)12 and T
(L)23 are given by the S-wave projected versions of Eq. (F.4) and
Eq. (F.6) without T(L)13 and without primed amplitudes T
′(L)ij . Before we write them down we
note that also the modified Kronecker-deltas are zero except for δ(12)P2P3
= 1 and hence we find
T(0)12 (E, k, p) =
2 π γ12 x1 f(3)(12)(12) S213
µ212 S12 c12
m1
kpQ122
0 (k, p;E)
+1
π
x2 f(3)(12)(12) S213
µ12 S12 c12
∫ ∞
0
dqq2 T
(0)12 (E, k, q) m1
qpQ122
0 (q, p;E)
−γ12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
+1
π
x4 f(3)(23)(12) S321
µ12
√S12 S23 c12 c23
∫ ∞
0
dqq2 T
(0)23 (E, k, q) m2
qpQ213
0 (q, p;E)
−γ23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
,
(5.10)
and
T(0)23 (E, k, p) =
2 π γ12 z1 f(1)(12)(23) S123
µ23 µ12
√S23 S12 c23 c12
m2
kpQ231
0 (k, p;E)
+1
π
z2 f(1)(12)(23) S123
µ23
√S23 S12 c23 c12
∫ ∞
0
dqq2 T
(0)12 (E, k, q) m2
qpQ231
0 (q, p;E)
−γ12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
.
(5.11)
93
The three particles P1, P2 and P3 are distinguishable bosons in the Zb–B∗ system. Hence, all
symmetry factors are equal to 1 independently of their indices. Using the summarized definitionson page 56 one obtains for the f and f factors
f(3)(12)(12) =
1
2, f
(3)(23)(12) =
1√2, f
(1)(12)(23) =
1√2. (5.12)
The remaining x and z parameters are spin and isospin dependent. Depending on the respectivechannel the results of appendix A.3 lead to the values summarized in Tab. 5.1. Since normalizedprojectors are considered in the mentioned method it is directly implied that all factors cij are
equal to one. One observes that in the spin 0 and 2 channels only the amplitude T(0)12 contributes
while for spin 1 both amplitudes are part of a coupled system. In the former case one finds
T(0)12 (E, k, p) =
−1
2
1
πγ m
µ2
1
2pkln
[k2
2µ+ p2
2µ′ − E + pkm− iε
k2
2µ+ p2
2µ′ − E −pkm− iε
]
+
−1
2
1
1
2π
m
µ
∫ ∞
0
dqq2 T
(0)12 (E, k, q)
−γ +
√−2µ
(E − q2
2m∗− q2
2(m+m∗)
)− iε
× 1
2pqln
[q2
2µ+ p2
2µ′ − E + pqm− iε
q2
2µ+ p2
2µ′ − E −pqm− iε
], (5.13)
where the upper components represent the spin 0 or spin 2 channel with isospin 1/2 and thelower ones are valid for spin 0 or spin 2 with isospin 3/2. Furthermore, the short-hand notationfor the Legendre function was replaced according to Eq. (3.73) and Eq. (D.10) as in the previoussection. The coupled integral equation system for the spin 1 channel is then given by
T(0)12 (E, k, p) =
12
−1
πγ m
µ2
1
2pkln
[k2
2µ+ p2
2µ′ − E + pkm− iε
k2
2µ+ p2
2µ′ − E −pkm− iε
]
+
12
−1
1
2π
m
µ
∫ ∞
0
dqq2 T
(0)12 (E, k, q)
−γ +
√−2µ
(E − q2
2m∗− q2
2(m+m∗)
)− iε
× 1
2pqln
[q2
2µ+ p2
2µ′ − E + pqm− iε
q2
2µ+ p2
2µ′ − E −pqm− iε
]
+
−1
2
1
1√2π
m∗µ
∫ ∞
0
dqq2 T
(0)23 (E, k, q)
−γ′ +√−2µ′
(E − q2
2m− q2
2(m∗+m∗)
)− iε
× 1
2pqln
[q2
2µ+ p2
2µ− E + pq
m∗− iε
q2
2µ+ p2
2µ− E − pq
m∗− iε
], (5.14)
94
T(0)23 (E, k, p) =
−1
2
1
√2πγ m∗µ′ µ
1
2pkln
[k2
2µ′ + p2
2µ− E + pk
m∗− iε
k2
2µ′ + p2
2µ− E − pk
m∗− iε
]
+
−1
2
1
1√2π
m∗µ′
∫ ∞
0
dqq2 T
(0)12 (E, k, q)
−γ +
√−2µ
(E − q2
2m∗− q2
2(m+m∗)
)− iε
× 1
2pqln
[q2
2µ′ + p2
2µ− E + pq
m∗− iε
q2
2µ′ + p2
2µ− E − pq
m∗− iε
], (5.15)
where again the upper entries must be used for isospin 1/2 and the lower ones for isospin 3/2.Moreover, it holds m1 = m, m2 = m3 = m∗, µ12 = µ, µ23 = µ′, γ12 = γ and γ23 = γ′ with thenotation introduced at the beginning of this section on elastic Z
(′)b –B(∗) scattering.
x1 x2 x4 z1 z2
spin 0 & isospin 1/2 −12−1
20 0 0
spin 0 & isospin 3/2 1 1 0 0 0
spin 1 & isospin 1/2 12
12−1
2−1
2−1
2
spin 1 & isospin 3/2 −1 −1 1 1 1
spin 2 & isospin 1/2 −12−1
20 0 0
spin 2 & isospin 3/2 1 1 0 0 0
Table 5.1: Spin and isospin channel dependent parameters for Zb–B∗ scattering.
5.1.3 Z ′b–B scattering amplitude
The three particle system P1 = B∗, P2 = B∗ and P3 = B again leads to a1 = b2 = a3 = 1, but itholds
δP1P2 = δP1P3 = δP2P3 = δA1A3 = δA2A3 = 0 ,
δA1A2 = 1 . (5.16)
This is not surprising since compared to the previous section on Zb–B∗ scattering we simply
interchanged the particle allocation of P1 and P2. Therefore one has to deal with the same twodimers
d12 = (η12 = +1) B∗B∗ , (5.17)
d23 =1√2
(B∗B + (η23 = +1)B∗B
), (5.18)
representing the Zb(10610) and the Z ′b(10650), but now d12 is interpreted as the latter. Againthere is no d13 = B∗B dimer and there are no primed ones at all in the system. However, in
95
contrast to the above discussed case there are less scattering channels because the spin 1⊗ 0 = 1is fixed. Applying δ
(12)P1P3
= δ(12)P2P3
= δ(23)P1P3
= 0 and δ(23)P1P2
= 1 to the corresponding amplitudes T(L)12
(Eq. (F.4)) and T(L)23 (Eq. (F.6)) and projecting both onto S-waves one ends up with
T(0)12 (E, k, p) =
1
π
x4 f(3)(23)(12) S321
µ12
√S12 S23 c12 c23
∫ ∞
0
dqq2 T
(L)23 (E, k, q) m2
qpQ213L (q, p;E)
−γ23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
,
(5.19)
and
T(0)23 (E, k, p) =
2 π γ12 z1 f(1)(12)(23) S123
µ23 µ12
√S23 S12 c23 c12
m2
kpQ231L (k, p;E)
+1
π
z2 f(1)(12)(23) S123
µ23
√S23 S12 c23 c12
∫ ∞
0
dqq2 T
(L)12 (E, k, q) m2
qpQ231L (q, p;E)
−γ12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
+1
π
z4 f(2)(23)(23) S231
µ23 S23 c23
∫ ∞
0
dqq2 T
(L)23 (E, k, q) m3
qpQ322L (q, p;E)
−γ23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
. (5.20)
The remaining parameters are according to their definitions on pages 56 and 57 given by
Sij = Sijk = 1 , ∀ i, j, k ∈ 1, 2, 3 ,
regarding the symmetry factors and by
f(3)(23)(12) =
1√2, f
(1)(12)(23) =
1√2, f
(1)(13)(23) =
1√2, f
(2)(23)(23) =
1
2.
Moreover, one needs the values given in Tab. 5.2 which are obtained using the methods presentedin appendix A.3, i.e. cij = 1 for all i < j ∈ 1, 2, 3.
x4 z1 z2 z4
spin 1 & isospin 1/2 12
12
12
12
spin 1 & isospin 3/2 −1 −1 −1 −1
Table 5.2: Spin and isospin channel dependent parameters for Z ′b–B scattering.
With the notation that upper entries are for isospin 1/2 and lower ones for isospin 3/2 this yieldsthe coupled integral equation system below (with m1 = m2 = m∗, m3 = m, µ12 = µ′, µ23 = µ,
96
γ12 = γ′ and γ23 = γ):
T(0)12 (E, k, p) =
12
−1
1√2π
m∗µ′
∫ ∞
0
dqq2 T
(L)23 (E, k, q)
−γ +
√−2µ
(E − q2
2m∗− q2
2(m∗+m)
)− iε
× 1
2pqln
[q2
2µ′ + p2
2µ− E + pq
m∗− iε
q2
2µ′ + p2
2µ− E − pq
m∗− iε
], (5.21)
T(0)23 (E, k, p) =
12
−1
√2πγ′ m∗µ µ′
1
2pkln
[k2
2µ+ p2
2µ− E + pk
m∗− iε
k2
2µ+ p2
2µ− E − pk
m∗− iε
]
+
12
−1
1√2π
m∗µ
∫ ∞
0
dqq2 T
(L)12 (E, k, q)
−γ′ +√−2µ′
(E − q2
2m− q2
2(m∗+m∗)
)− iε
× 1
2pqln
[q2
2µ′ + p2
2µ− E + pq
m∗− iε
q2
2µ′ + p2
2µ− E − pq
m∗− iε
]
+
12
−1
1
2π
m
µ
∫ ∞
0
dqq2 T
(L)23 (E, k, q)
−γ +
√−2µ
(E − q2
2m∗− q2
2(m∗+m)
)− iε
× 1
2pqln
[q2
2µ+ p2
2µ′ − E + pqm− iε
q2
2µ+ p2
2µ′ − E −pqm− iε
], (5.22)
where it was again used that one can – according to Eq. (3.73) and Eq. (D.10) – write the L = 0Legendre function of the second kind in terms of the logarithm.
5.1.4 Z ′b–B∗ scattering amplitude
Finally, we consider Z ′b–B∗ scattering leading to the three particle system P1 = P3 = B∗ and
P2 = B∗ which corresponds to a1 = b2 = a3 = 1 and
δP1P2 = δP2P3 = 0 ,
δP1P3 = δA1A2 = δA1A3 = δA2A3 = 1 . (5.23)
From Eq. (3.13) it follows – keeping in mind that there is up to now no experimental evidencefor a B∗B∗ state with large scattering length – that there is just one dimer d12 = η12B
∗B∗ in thesystem (cf. rules on page 51) whose G-parity is according to Tab. 4.1 given by η12 = +1. The
corresponding S-wave three-body amplitude T(0)12 is defined by Eq. (F.4) where one has erased
97
all contributions from the other amplitudes:
T(0)12 (E, k, p) =
2 π γ12
µ212 S12 c12
×[x1 δ
(12)P1P3
f(3)(12)(12) S123
m2
kpQ211L (k, p;E) + x1 δ
(12)P2P3
f(3)(12)(12) S213
m1
kpQ122L (k, p;E)
]
+1
π
1
µ12 S12 c12
∫ ∞
0
dqq2 T
(L)12 (E, k, q)
−γ12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
×[x2 δ
(12)P1P3
f(3)(12)(12) S123
m2
qpQ211L (q, p;E) + x2 δ
(12)P2P3
f(3)(12)(12) S213
m1
qpQ122L (q, p;E)
].
(5.24)
From the summarized definitions on pages 56 and 57 one deduces δ(12)P1P3
= 1, but δ(12)P2P3
= 0 and
S12 = S123 = 1 ,
S123 = 2 ,
f(3)(12)(12) = f
(3)(12)(12) = 1 .
With Eqs. (A.111, A.129) derived in appendix A.3 for normalized projection operators (i.e.c12 = 1) one additionally finds the values shown in Tab. 5.3 for the spin and isospin dependent xfactors. Hence, one ends up with the following amplitude (with m1 = m2 = m3 = m∗, µ12 = µ′
and γ12 = γ′ as explained at the beginning of section 5.1):
T(0)12 (E, k, p) =
12
−1−1
412
−14
12
2πγ′ m∗µ′2
1
2pkln
[k2
2µ′ + p2
2µ′ − E + pkm∗− iε
k2
2µ′ + p2
2µ′ − E −pkm∗− iε
]
+
12
−1−1
412
−14
12
1
π
m∗µ′
∫ ∞
0
dqq2 T
(L)12 (E, k, q)
−γ′ +√−2µ′
(E − q2
2m∗− q2
2(m∗+m∗)
)− iε
× 1
2pqln
[q2
2µ′ + p2
2µ′ − E + pqm∗− iε
q2
2µ′ + p2
2µ′ − E −pqm∗− iε
], (5.25)
where the entries in curly brackets represent the channels from top to bottom in the same orderas in Tab. 5.3 and where Eq. (3.73) together with Eq. (D.10) were used to write the Legendrefunctions in terms of the logarithm.
98
x1 x2 x1 x2
spin 0 & isospin 1/2 12
12
12
12
spin 0 & isospin 3/2 −1 −1 −1 −1
spin 1 & isospin 1/2 −14−1
4−1
4−1
4
spin 1 & isospin 3/2 12
12
12
12
spin 2 & isospin 1/2 −14−1
4−1
4−1
4
spin 2 & isospin 3/2 12
12
12
12
Table 5.3: Spin and isospin channel dependent parameters for Z ′b–B∗ scattering.
5.2 Numerical determination of scattering length and
phase shift
In this section it will be described how the three-body observables scattering length and phaseshift can be deduced from the discretized scattering amplitude, given in form of a matrix equationT = R+MT. The discretization itself is explained in appendix G. For a single integral equationdescribing Z
(′)b –B(∗) scattering in an arbitrary channel there are three relevant regions of the
center-of-mass energy E (the binding energy of the relevant molecule Zb or Z ′b is denoted asB(Z) which is equal to either B or B′):
• −B(Z) ≤ E ≤ 0: in terms of the momentum k this energy region translates to 0 ≤ k ≤kmax where kmax is the root of E(k) which defines the momentum where the moleculebreaks apart into its constituents. Hence, in this region the elastic scattering of abottom meson off a molecule takes place.
• −∞ < E < −B(Z): for smaller energies below the molecule threshold there would appear– if present – three particle bound states due to the Efimov effect.
• 0 < E < ∞: for larger energies the molecule will break apart and one has to deal witha system of three individual bottom mesons which scatter off each other. However,this system will not be further discussed in this work.
In a system of two coupled integral equations where both Zb and Z ′b are involved one has toreplace in the second case −∞ < E < −B(Z) by −∞ < E < −max(B,B′) as a trimer statemust lie below both dimer thresholds. In the first case one has to take care of the relation betweenthe two binding energies B and B′. The elastic two-body scattering Zb–B
∗ only takes place forB ≥ B′, namely in the energy region −B ≤ E ≤ −B′. In the other case, i.e. for B < B′ bothmolecular states can be built of the three bottom mesons. Thus, there is no region where wewould deal with a two-body problem alone. Since there is no straightforward way of describingsuch a three-body scattering problem in the used theory we will not analyze it in more detail.The second coupled system appears in Z ′b–B scattering. Here, the situation is interchanged: forB < B′ the valid energy region is −B′ ≤ E ≤ −B and the inaccessible case is given by B ≥ B′.
99
5.2.1 Method
As we know already from section 4.1 that there is no Efimov effect in Z(′)b –B(∗) scattering we
can restrict the discussion to the elastic molecule–particle scattering region. According to thedimer auxiliary field trick (cf. section 2.1) the molecule is treated as one particle and henceone can describe molecule–particle scattering in terms of two-body scattering theory which wasdiscussed in section 1.4. Consequently, one can write for the elastic S-wave amplitude T
(0)12 in all
four processes:
T(0)12 (k) =
2π
µ(12)3
1
k cot δ − ik , (5.26)
with leading order effective range expansion
T(0)12 (k) =
2π
µ(12)3
1
− 1a3− ik . (5.27)
From the considerations above it is clear that one needs the explicit form of the amplitudeT
(0)12 (E, k, p). It can be found by numerically solving the inhomogeneous matrix equation T =
R +MT (see appendix G) for a given momentum k, i.e. for a given center-of-mass energy
E ∼ k2. Here, the vector T is given by T = T(0)12 for the single integral equations of Zb–B
and Z ′b–B∗ scattering and by T =
(T
(0)12 , T
(0)23
)Tfor the coupled systems of Zb–B
∗ and Z ′b–B
scattering (cf. appendix G). These matrix equations are solved using the ZGESV routine of theLAPACK library which is documented in Ref. [190].Thus, one concludes from Eq. (5.27) that the three-body molecule–particle scattering length a3
can be determined via
a3 = −µ(12)3
2πRe[T
(0)12 (0)
]. (5.28)
In a numerical calculation one sets k = p1 (i.e. k is equal to the first mesh point, cf. appendix G)
instead of exactly zero and uses T(0)12 (k) ≡ T
(0)12 (E = k2/(2µ(12)3), k, p = k). This approach is
justified since for a typical number of 100 mesh points one gets p1 < 0.005 even for large cutoffsΛ ∼ 10000 MeV. Therefore p1 is sufficiently close to 0 and hence no extrapolation is needed.The second observable, namely the scattering phase shift, can be determined as a function of themomentum k by rearranging Eq. (5.26):
δ(k) = arccot
(2π
µ(12)3
1
kRe
[1
T(0)12 (k)
]). (5.29)
Note, that for each k one has to take – after solving the matrix equation – that part of theamplitude T
(0)12 (k) which corresponds to the pole contribution, that is, in the discretized version
TN−1 or TN−2 depending on the number of poles in the scattering process (cf. appendix G).Besides the predictions one can use Eq. (5.26) in the form
− 2π
µ(12)3
Im
[1
T (k)
]= k , (5.30)
as a check for the numerics used, in the sense that a linear dependence is from a numerical pointof view very sensitive to any programming error.
100
5.2.2 Results
The (coupled) integral equations derived in section 5.1 will now be analyzed using the numericalmethods explained in the previous section and in appendix G. The binding energies needed asinput for these calculations are obtained by Cleven et al. via an analysis of bottom meson loopsin the framework of hadronic molecules which is presented in Ref. [139]. The values are:
B = 4.7+2.3−2.2 MeV , (5.31)
B′ = 0.11+0.14−0.06 MeV , (5.32)
which lead – according to Eq. (1.22) – to the binding momenta
γ = 157.9+38.6−37.0 MeV , (5.33)
γ′ = 24.20+15.40−6.60 MeV . (5.34)
Hence, one observes that the value for the Zb(10610) is larger than the pion mass which inprinciple contradicts the application of EFT(/π), however, due to the large uncertainties of γ itis not excluded that the binding momentum of Zb is below mπ (though close to it) so that onecan use a pionless EFT at least to obtain some first insights.
Discussion of Zb–B scattering
As a consequence of the fact that there is no Efimov effect in the system, the elastic scatteringZb–B is completely described by the formulae in section 5.2.1 which lead to cutoff independentvalues for the two observables, Zb–B scattering length a3 and S-wave phase shift δ(k). Thescattering length in the I = 3/2 & S = 1 channel is given by
aI= 3
2,S=1
3 = −15.13 fm , (5.35)
and the corresponding phase shift in this channel is shown as a function of k in Fig. 5.1. As itis known from basic scattering theory [81,82] a positive phase shift corresponds to an attractivepotential between the two scattering particles. However, such a potential can only induce a
S-wave bound state if the scattering length is also positive. Since aI= 3
2,S=1
3 is negative thisobservation agrees with the absence of the Efimov effect in the system. From Fig. 5.1 we concludethat the scattering length in the other channel with I = 1/2 & S = 1 must be positive since thephase shift is negative. This is indeed the case:
aI= 1
2,S=1
3 = 0.62 fm . (5.36)
Discussion of Zb–B∗ scattering
In the same way as for Zb–B scattering one can analyze the coupled Zb–B∗ system. Again one
can predict the scattering length and the phase shift in all six isospin-spin channels of the elasticscattering process. Since the projection onto some of the isospin and spin states yields identical
101
0 50 100 150 200k / MeV
-20
0
20
40
60δ
/ deg
I = 3/2, S = 1 channelI = 1/2, S = 1 channel
Figure 5.1: S-wave phase shift δ as function of the momentum k for both channels in elasticZb–B scattering.
prefactors, there only remain four different values and curves. The latter are shown in Fig. 5.2and the scattering lengths are given by
aI= 3
2,S=0,2
3 = −15.66 fm , (5.37)
aI= 1
2,S=0,2
3 = 0.62 fm , (5.38)
aI= 3
2,S=1
3 = 0.88 fm , (5.39)
aI= 1
2,S=1
3 = −1.97 fm . (5.40)
Thus, the correlation between the signs of δ and a3 agrees with the missing Efimov trimers.
Discussion of Z ′b–B scattering
It is already known that there is no Efimov effect in the Z ′b–B system. But moreover, it holdsB ' 4.7 MeV ≥ 0.1 MeV ' B′. Thus, it is not possible (according to section 5.2.1) to extractother observables for this process. A different approach would be necessary to deal with such anon-trivial three-body system which is beyond the aim of this work.
102
0 50 100 150 200k / MeV
-20
0
20
40
60
δ / d
eg
I = 3/2, S = 0,2 channelI = 1/2, S = 0,2 channelI = 3/2, S = 1 channelI = 1/2, S = 1 channel
Figure 5.2: S-wave phase shift δ as function of the momentum k for all six channels in elasticZb–B
∗ scattering. Note, that the S = 0 and S = 2 spin channels yield the same result.
Discussion of Z ′b–B∗ scattering
Finally, the analysis of Z ′b–B∗ scattering allows us to obtain the scattering lengths and the phase
shifts which are shown in Fig. 5.3 and given by
aI= 3
2,S=1,2
3 = aI= 1
2,S=0
3 = −99.02 fm , (5.41)
aI= 1
2,S=1,2
3 = 4.03 fm , (5.42)
aI= 3
2,S=0
3 = 9.61 fm . (5.43)
One observes that the absolute value of the scattering length in the I = 3/2 & S = 1, 2 and inthe I = 1/2 & S = 0 channel is almost two orders of magnitude larger than in all other processesand channels. In Ref. [82] it is explained that for a growing attractive potential, the scatteringlength a3 tends to minus infinity until it appears a bound state in the system which causes atB3 = 0, that a3 changes its sign to positive infinity. Hence, for B3 > 0, a3 is positive and finite.Consequently, the large absolute value of the scattering length indicates that the interactionbetween Z ′b and B∗ is just quite not strong enough to from a bound state.
103
0 5 10 15 20 25 30k / MeV
-60
-40
-20
0
20
40
60
δ / d
eg
I = 3/2, S = 1,2 & I = 1/2, S = 0 channelI = 3/2, S = 0 channelI = 1/2, S = 1,2 channel
Figure 5.3: S-wave phase shift δ as function of the momentum k for all six channels in elasticZ ′b–B
∗ scattering. Note, that for each isospin state the S = 1 and S = 2 spin channels yield thesame result and additionally that the I = 1/2 & S = 0 result is equivalent to that of I = 3/2 &S = 1, 2.
104
Chapter 6
Conclusion and outlook
In this work we considered the generic three-body scattering of a particle off a S-wave dimer in apionless non-relativistic effective field theory. This was done in order to derive a transcendentalequation whose solution tells us whether or not the corresponding three-body system is affectedby Efimov physics. For this purpose we made the assumption that the constituents of the two-body bound or virtual state are heavy compared to pions, but that their binding momentumis rather small, i.e. at most of the order of the pion mass, so that EFT(/π) is applicable. Themethod we have used to search for Efimov trimers in a three particle system was to decouplethe molecule–particle scattering amplitudes within the coupled integral equation system in thelimit of asymptotic large momenta, that is, to diagonalize the matrix whose elements containall information about spin, isospin, flavor normalization factors and symmetry factors. Unfor-tunately, the decoupling / diagonalizing is only possible if one imposes some constraints on themasses and the existence of shallow dimer states between some particles. We found two types ofsystems which are suitable for this method: type 1 must have (approximately) equal masses (butno further constraints) and type 2 must have m2 = m3 or m1 = m3 (remember the freedom inthe allocation of particles 1 and 2) with the restriction that the equal mass particles must haveno bound or virtual state (i.e. they do not have a large scattering length). Only systems madeof two equal mass particles with a dimer and a third particle of different mass and in additionsystems where all particles have different masses cannot be described by the presented method.However, especially in a system with three unequal masses whose difference is so large that eventhe approximation of equal masses is not justified, it is relatively unlikely that such particles atall have a bound state explained by the strong force. Besides these subtleties there are no moreconstraints. Hence, one finally finds a transcendental equation for the scaling parameter s whichcan have – depending on the eigenvalues of the matrix and on the mass ratios – a purely imaginarysolution which corresponds to the existence of the Efimov effect. Although this method couldbe used in many different fields of few-body physics (e.g. cold atoms or halo nuclei) we appliedit to hadronic molecules scattering off a third particle which has a large scattering length withat least one of the constituents of the molecule. Indeed, we found that a couple of establishedparticles – interpreted as hadronic molecules – are affected by Efimov physics. In particular, thethree kaon system KKK has a reasonably small scaling factor of 1986.14. In the up to nowknown charm and bottom meson system no Efimov effect was found. Instead, we have analyzedthe – due to the absence of Efimov trimers – elastic Z
(′)b –B(∗) scattering as a selected example in
105
order to obtain three-body observables like S-wave scattering length and phase shift in systemswithout Efimov effect using EFT(/π). We have also checked that hypothetical molecules made ofidentical bosons (with spin and isospin) or fermions have for most spin and isospin configurationsan intermediate Efimov trimer state in their scattering off a third identical boson or fermion.Although it might be experimentally challenging to find such new states it might also be worththe effort because the existence of trimer states would be hard to explain in the tightly bounddiquark picture, but would emerge naturally in the hadronic molecule interpretation. Hence, inthis sense this work provides in principle a further method to clarify the nature of – at least some– exotic particles in the charmonium and bottomonium sector.As an outlook one can say that the extension of this work to P -wave or even higher L dimerstates would be a nice feature, especially in cold atoms or halo nuclei where the existence ofhigher partial wave dimers is more common than in the hadronic molecule sector. Furthermore,an inclusion of higher order terms would obviously improve the quality of the results. However, itwould require more experimental knowledge about the scattering length and the effective range ofthe particles. Also the extension of the used pionless EFT to a EFT with explicit pions (XEFT)would be useful, especially for the not so shallowly bound molecular states. Independently ofthe theoretical effort it is expected that the planned or already running experiments like Belle II,BES III or LHCb will provide more data which could help improving the theoretical predictions.
106
Appendix A
Spin and isospin projection operators
In the first part of the appendix our goal is to explain the structure of the combined spin andisospin projection operators Oij. In the second part we focus on the projection onto a specificscattering channel in order to derive the defining equations for the parameters x, y and z. Finally,in the last subsection we will discuss the relation between these parameters and the Wigner 6-Jsymbol which gives us the opportunity to determine them for arbitrary spin and isospin withoutexplicit knowledge of the projection operators. Since spin and isospin are described equivalentlywe derive all relations only considering spin. To generalize the results to isospin is straightforward(in fact, it corresponds to simply renaming all variables).
A.1 Combined spin and isospin projection operators
We start with some conventions regarding the matrix elements defining a generic scatteringamplitude T ∼ 〈out| ... |in〉 with ket and bra vectors defined according to the conventions ofRef. [1]:
〈k,p| ∼ 〈0| apak ,|k,p〉 ∼ a†ka
†p |0〉 . (A.1)
Hence, it holds
(〈k,p|)† = (〈0| apak)† = a†ka†p |0〉 = |k,p〉 . (A.2)
Here, ap is the annihilation and a†p is the creation operator which annihilates and creates a statewith momentum p, respectively. Again following the conventions of Ref. [1] the field used in theLagrangian density A is related to a and A† to a† by a Fourier transform. Thus, we will use thefields A and A† as synonyms for annihilation and creation operators, respectively. Since we areworking in a non-relativistic theory it holds that the contraction of a state with momentum pand the operator A yields 1, i.e.
〈p|A† = 1 ,
A |p〉 = 1 . (A.3)
107
= i T˜β γα (k,p,q)
(Ai)˜β
(Aj)γ
α
˜β
γ
(dij)αk p
q
Figure A.1: Diagrammatic representation of a generic dimer decay. The corresponding ampli-
tude Tβ γ
α depends on the incoming and outgoing spin indices.
Adding spin degrees of freedom to the theory the fields Aα become proportional to polarizationvectors (~εα)
αfor which we use the same symbol independently of the exact spin. In fact, for
α = being a scalar spin ”index“ the polarization vector becomes ~εα = 1. For α = α ∈ 1, 2being a spin 1/2 index one gets a non-relativistic two dimensional spinor ~εα = ~χα and forα = i ∈ 1, 2, 3 being a spin 1 index ~εα = ~εi is the usual polarization vector of vector particles.The contractions in Eq. (A.3) are thus changed to contractions of a state with momentum p andspin α with the operator Aα where α denotes the component of the polarization vector ~εα:
〈p, α|A†α =(~ε †α
)α
(p) , (A.4)
Aα |p, α〉 = (~εα)α
(p) . (A.5)
To clarify the notation consider a state 〈p, (i = 1)| which is contracted with an operator A†i whichis associated to a spin 1 field:
〈p, (i = 1)|A†i = (~ε1)i (p) =
100
i
=
1 , for i = 1
0 , for i = 2
0 , for i = 3
, (A.6)
where we have used the Cartesian polarization basis.
After this introduction we can continue and use the relations above to derive the projectionoperators Oij which appear in the Lagrangian density Eq. (3.23). Note, that the results forprojectors up to spin 1⊗ 1 are collected in Tab. A.1 in section A.4. In order to proof that theseprojectors Oij in Tab. A.1 indeed are correct we consider the vertex function of a dimer decayinginto its constituents which is in Fig. A.1 shown as a Feynman diagram (note, that we can restrictourselves without loss of generality to the case dij ∼ AjAi). We will compare the numerical valueof the decay amplitude with the corresponding Clebsch-Gordan coefficient. If we ignore for themoment the non-trivial wave function renormalization of the dimer field dij the decay amplitudeis according to the vertex functions in Fig 3.1 given by
Tβ γ
α (k,p,q) =⟨q, β; p, γ
∣∣∣ aiaj(−gij)(A†i
)β
(Oij)α,βγ(A†j
)γ
(dij)α
∣∣∣k, α⟩
= −gij ai aj(~ε †β
)
β
(Oij)α,βγ(~ε †γ
)γ
(~εα)αδ(3)(p + q− k) , (A.7)
108
where we have in the second step contracted the operators with the external lines according toFig. A.1. If we additionally set the momenta to k = p+q the δ-function yields 1 and we concludefrom phenomenology that the expression
C :=
(~ε †β
)
β
(Oij)α,βγ(~ε †γ
)γ
(~εα)α, (A.8)
as part of the amplitude Tβ γ
α (p,q) must be equal to the Clebsch-Gordan coefficient for coupling
spin β of Ai and spin γ of Aj to a total spin α of dij. In the following we will check this conditionexemplary for the spin coupling 1/2 ⊗ 1 → 1/2, but first we give some remarks regarding thenotation.We write all projectors in terms of non-relativistic spinor indices α, β, γ, ... ∈ 1, 2 and in termsof the Cartesian polarization vector indices i, j, k, ... ∈ 1, 2, 3 which describe spin 1/2 and spin1 interactions, respectively. The underlying group of spin 1/2 is SU(2) whose generators are thethree Pauli-matrices
σ1 =
(0 11 0
), σ2 =
(0 −ii 0
), σ3 =
(1 00 −1
), (A.9)
and for spin 1 it is the three dimensional rotation group SO(3) which is generated by the matrices
U1 =
0 0 00 0 −i0 i 0
, U2 =
0 0 i0 0 0−i 0 0
, U3 =
0 −i 0i 0 00 0 0
. (A.10)
For both groups the generators fulfill besides other properties the following useful relations. Forthe Pauli-matrices it holds:
σ2i = 1 ∀ i ,
Tr(σi) = 0 ∀ i ,Tr(σiσj) = 2 δij ∀ i, j ,
σiσj = δij 1+ i εijk σk ∀ i, j ,[σi, σj] = 2i εijk σk ∀ i, j ,σi, σj = 2δij 1 ∀ i, j . (A.11)
And similarly the matrices Ui fulfill:
(Ui)jk ≡ −i εijk ∀ i, j, k ,Tr(Ui) = 0 ∀ i ,
Tr(UiUj) = 2 δij ∀ i, j ,(UiUj)k` = δij δk` − δi`δkj ∀ i, j, k, ` ,[Ui, Uj] = i εijk Uk ∀ i, j . (A.12)
109
The spin indices we are working with are given in the Cartesian polarization basis, namely, theycorrespond to the spin 1/2 polarization vectors ~χα and the spin 1 polarization vectors ~εi givenby
~χ1 =
(10
), ~χ2 =
(01
)and
~ε1 =
100
, ~ε2 =
010
, ~ε3 =
001
, (A.13)
respectively. Since ~χ1,2 are the eigenvectors to the eigenvalues λ1,2 = ±1/2 of the matrix σ3/2one can directly identify ~χα with the physical spin states defined by their magnetic quantumnumber m:
~χ1 =m = +1
2,
~χ2 =m = −1
2. (A.14)
In the spin 1 case one has to perform a change of basis from Cartesian (~ε1, ~ε2, ~ε3) to spherical(~ε+, ~ε0, ~ε−) polarization vectors where the momentum p points in z-direction. The latter canbe identified with physical spin states m = 0, ±1 in the following manner:
~ε+ = − 1√2
(~ε1 + i~ε2) = − 1√2
1i0
=m = +1 ,
~ε0 = ~ε3 =
001
=m = 0 ,
~ε− =1√2
(~ε1 − i~ε2) =1√2
1−i0
=m = −1 , (A.15)
meaning that ~ε+, ~ε0 and ~ε− are the normalized eigenvectors to the eigenvalues +1, 0 and −1 ofthe generator U3. Alternatively, one can maintain the Cartesian basis indices in the Lagrangiandensity and insert
j = − 1√2
(1 + i2) for a m = +1 state,
j =1√2
(1− i2) for a m = −1 state,
j = 3 for a m = 0 state, (A.16)
instead (note, that the i on the right-hand-side is the imaginary unit while j on the left is a spinindex). This is the method we will use.
110
Ket spin vectors |j,m〉 are then related to the polarization vectors via
~χα =
∣∣∣∣1
2,1
2
⟩, for α = 1 ,
~χα =
∣∣∣∣1
2,−1
2
⟩, for α = 2 ,
~εi = |1, 1〉 , for i = − 1√2(1 + i2) ,
~εi = |1, 0〉 , for i = 3 ,
~εi = |1,−1〉 , for i = − 1√2(1− i2) . (A.17)
From the Clebsch-Gordan coefficients it additionally follows that one can identify for spin 3/2the ket vectors with
~εi ~χα =
∣∣∣∣3
2,3
2
⟩, for
i = − 1√
2(1 + i2), α = 1
,
~εi ~χα =
∣∣∣∣3
2,1
2
⟩, for 1√
3
i = − 1√
2(1 + i2), α = 2
+√
2√3i = 3, α = 1 ,
~εi ~χα =
∣∣∣∣3
2,−1
2
⟩, for
√2√3i = 3, α = 2+ 1√
3
i = 1√
2(1− i2), α = 1
,
~εi ~χα =
∣∣∣∣3
2,−3
2
⟩, for
i = 1√
2(1− i2), α = 2
, (A.18)
and similarly for spin 2:
~εi ~εj = |2, 2〉 , fori = − 1√
2(1 + i2), j = − 1√
2(1 + i2)
,
~εi ~εj = |2, 1〉 , for 1√2
i = − 1√
2(1 + i2), j = 3
+ 1√
2
i = 3, j = − 1√
2(1 + i2)
,
~εi ~εj = |2, 0〉 , for 1√6
i = − 1√
2(1 + i2), j = 1√
2(1− i2)
+√
2√3
i = 3, j = 3
,
+ 1√6
i = 1√
2(1− i2), j = − 1√
2(1 + i2)
,
~εi ~εj = |2,−1〉 , for 1√2
i = 3, j = 1√
2(1− i2)
+ 1√
2
i = 1√
2(1− i2), j = 3
,
~εi ~εj = |2,−2〉 , fori = 1√
2(1− i2), j = 1√
2(1− i2)
. (A.19)
The corresponding bra spin vectors 〈j,m| = (|j,m〉)† can be found by Hermitian conjugation ofthe relations above.
Let us now consider the coupling(J(Ai) = 1/2
)⊗(J(Aj) = 1
)→(J(dij) = 1/2
). From
Tab. A.1 we take the corresponding projection operator
(Oij)α,βγ =1√3
(σg)βα ,
111
where α = α and β = β are spin 1/2 indices and γ = g is a spin 1 index. Plugging this intoEq. (A.8) we find
C =(~χ †β
)β
1√3
(σg)αβ
(~ε †g
)g
(~χα)α
=1√3
[(~χ †β
)1
((~ε †g
)1− i(~ε †g
)2
)(~χα)
2+(~χ †β
)2
((~ε †g
)1
+ i(~ε †g
)2
)(~χα)
1
+(~χ †β
)1
(~ε †g
)3
(~χα)1−(~χ †β
)2
(~ε †g
)3
(~χα)2
]. (A.20)
Depending on α, β and g the equation above yields
• for α = 1, β = 1:
C =1√3
(~ε †g
)3
=
0 , for g = − 1√2(1− i2)
1√3, for g = 3
0 , for g = 12(1 + i2)
, (A.21)
• for α = 2, β = 1:
C =1√3
((~ε †g
)1− i(~ε †g
)2
)=
1√3
(− 1√
2− i i√
2
)= 0 , for g = − 1√
2(1− i2)
0 , for g = 31√3
(1√2− i i√
2
)=√
2√3, for g = 1
2(1 + i2)
,
(A.22)
• for α = 1, β = 2:
C =1√3
((~ε †g
)1
+ i(~ε †g
)2
)=
1√3
(− 1√
2+ i i√
2
)= −
√2√3, for g = − 1√
2(1− i2)
0 , for g = 31√3
(1√2
+ i i√2
)= 0 , for g = 1
2(1 + i2)
,
(A.23)
• for α = 2, β = 2:
C = − 1√3
(~ε †g
)3
=
0 , for g = − 1√2(1− i2)
− 1√3, for g = 3
0 , for g = 12(1 + i2)
. (A.24)
112
These results can be compared with the Clebsch-Gordan coefficients in Ref. [2]:( ⟨
12, 1
2
∣∣⊗ 〈1, 1|) ∣∣1
2, 1; 1
2, 1
2
⟩= (−1)
12−1−1
2
(〈1, 1| ⊗
⟨12, 1
2
∣∣) ∣∣1, 1
2; 1
2, 1
2
⟩= 0 ,
( ⟨12, 1
2
∣∣⊗ 〈1, 0|) ∣∣1
2, 1; 1
2, 1
2
⟩= (−1)
12−1−1
2
(〈1, 0| ⊗
⟨12, 1
2
∣∣) ∣∣1, 1
2; 1
2, 1
2
⟩= 1√
3,
( ⟨12, 1
2
∣∣⊗ 〈1,−1|) ∣∣1
2, 1; 1
2, 1
2
⟩= (−1)
12−1−1
2
(〈1,−1| ⊗
⟨12, 1
2
∣∣) ∣∣1, 1
2; 1
2, 1
2
⟩= 0 , (A.25)
( ⟨12, 1
2
∣∣⊗ 〈1, 1|) ∣∣1
2, 1; 1
2,−1
2
⟩= (−1)
12−1−1
2
(〈1, 1| ⊗
⟨12, 1
2
∣∣) ∣∣1, 1
2; 1
2,−1
2
⟩= 0 ,
( ⟨12, 1
2
∣∣⊗ 〈1, 0|) ∣∣1
2, 1; 1
2,−1
2
⟩= (−1)
12−1−1
2
(〈1, 0| ⊗
⟨12, 1
2
∣∣) ∣∣1, 1
2; 1
2,−1
2
⟩= 0 ,
( ⟨12, 1
2
∣∣⊗ 〈1,−1|) ∣∣1
2, 1; 1
2,−1
2
⟩= (−1)
12−1−1
2
(〈1,−1| ⊗
⟨12, 1
2
∣∣) ∣∣1, 1
2; 1
2,−1
2
⟩=√
2√3, (A.26)
( ⟨12,−1
2
∣∣⊗ 〈1, 1|) ∣∣1
2, 1; 1
2, 1
2
⟩= (−1)
12−1−1
2
(〈1, 1| ⊗
⟨12,−1
2
∣∣) ∣∣1, 1
2; 1
2, 1
2
⟩= −
√2√3,
( ⟨12,−1
2
∣∣⊗ 〈1, 0|) ∣∣1
2, 1; 1
2, 1
2
⟩= (−1)
12−1−1
2
(〈1, 0| ⊗
⟨12,−1
2
∣∣) ∣∣1, 1
2; 1
2, 1
2
⟩= 0 ,
( ⟨12,−1
2
∣∣⊗ 〈1,−1|) ∣∣1
2, 1; 1
2, 1
2
⟩= (−1)
12−1−1
2
(〈1,−1| ⊗
⟨12,−1
2
∣∣) ∣∣1, 1
2; 1
2, 1
2
⟩= 0 , (A.27)
( ⟨12,−1
2
∣∣⊗ 〈1, 1|) ∣∣1
2, 1; 1
2,−1
2
⟩= (−1)
12−1−1
2
(〈1, 1| ⊗
⟨12,−1
2
∣∣) ∣∣1, 1
2; 1
2,−1
2
⟩= 0 ,
( ⟨12,−1
2
∣∣⊗ 〈1, 0|) ∣∣1
2, 1; 1
2,−1
2
⟩= (−1)
12−1−1
2
(〈1, 0| ⊗
⟨12,−1
2
∣∣) ∣∣1, 1
2; 1
2,−1
2
⟩= − 1√
3,
( ⟨12,−1
2
∣∣⊗ 〈1,−1|) ∣∣1
2, 1; 1
2,−1
2
⟩= (−1)
12−1−1
2
(〈1,−1| ⊗
⟨12,−1
2
∣∣) ∣∣1, 1
2; 1
2,−1
2
⟩= 0 , (A.28)
where it was used that(〈j1,m1| ⊗ 〈j2,m2|
)|j1, j2; J,M〉 = (−1)J−j1−j2
(〈j2,m2| ⊗ 〈j1,m1|
)|j2, j1; J,M〉 (A.29)
holds with the standard notation of Ref. [2]. Indeed, one observes that C is equivalent to theClebsch-Gordan coefficient of the respective coupling. In the same way one could proof that theother projectors given in Tab. A.1 are correct.
A.2 Projection onto scattering channel
Up to now the decay amplitude Eq. (A.7) still has free (underlined) indices. In fact, β and γrepresent two polarization vectors ~εβ and ~εγ which can be coupled to – in general – more then
one total spin called ρ. Hence, the decay amplitude with initial spin state η and final state spinρ can be written as
Tρ
η =1√c
(O†T)ρ,γβ
Tβγ
α δαη , (A.30)
113
where we have introduced an operator OT which projects onto a specific final state ρ, i.e. which
couples the spins β and γ to a total spin ρ. Since this projector must be normalized in order to donot over-count some spin states, there is an normalization factor of 1/
√c added. The Kronecker-
delta δαη in Eq. (A.30) is nothing else than the (already normalized) projection operator whichcouples ”nothing“ and spin α to spin η. To get the full index-free decay amplitude one has – inthe usual way, cf. Ref. [1] – to average over initial and sum over final state spins leading to
T =1
dof(η)
∑
η,ρ
Tρ
η =1
dof(η)
∑
η,ρ
1√c
(O†T)ρ,γβ
Tβγ
α δαη . (A.31)
We now introduce a convention for scattering amplitudes, namely that initial spins are writtenas subscript and final state spins as superscript. Next, we consider a generic two-body scattering
amplitude Tγσ
αβand know from the considerations above that the full amplitude T is given by
T =1
dof(η)
∑
η,λ
T λη =1
dof(η)
∑
η,λ
1√cc′
(O†T)λ,σγ
Tγσ
αβ(O′T )η,αβ , (A.32)
with η being the total initial and λ being the total final state spin onto which the two operators
O′T and OT project. Note, that for elastic scattering one has to set η = λ as initial and final
state must be equal in this case. The projection operators O(′)T have the same structure as the
operators O derived above since both couple two spins to a total spin. The only difference is thatthe former couple the dimer spin and the spin of the third particle to the scattering channel spinwhile the latter couple the spins of the two constituent particles to the spin of the correspondingdimer. It is thus not necessary to give OT explicitly; it can be deduced from Tab. A.1.In the derivation of the transcendental equation for different types of systems (cf. section 3)
the deduced scattering amplitudes on the one hand have the same index structure (Tij)γσ
αβas the
generic one in Eq. (A.32). On the other hand they are given as integral equation, that is, theyare proportional to themselves and to the other amplitudes Tik and Tjk, but with different finalstate indices. To explain the method of projecting onto a specific spin channel it is sufficient toconsider only one example as which we chose the T23(q) contribution to the T13(p) amplitude (cf.second row of the first diagram on page 158):
(T13)γσ
αβ∼⟨p, γ;−p, σ
∣∣∣(d†13
)γ
(A3)ρ
(O†13
)γ,ρν
(A1)ν (T23)µναβ
×(A†2
)σ
(O23)µ,σρ
(A†3
)ρ
(d23)µ
∣∣∣k, α;−k, β⟩
=(~ε †γ
)γ
(~ε †σ
)σ
(O†13
)γ,ρν
(T23)µναβ
(O23)µ,σρ (~εα)α
(~εβ
)β
= δγγ δσσ
(O†13
)γ,ρν
(T23)µναβ
(O23)µ,σρ δαα δββ
=(O†13
)γ,ρν
(T23)µν
αβ(O23)µ,σρ , (A.33)
114
where we have used in the third step that in Cartesian spin basis (~εα)α
= δαα holds for all αwhich directly follows from the definition in Eq. (A.13). The full index-free amplitude thereforehas the proportionality
T13 =1
dof(η)
∑
η,λ
(T13)λη ∼1
dof(η)
∑
η,λ
1√cT12cT13
(O†T13
)λ,σγ
(O†13
)γ,ρν
(T23)µν
αβ(O23)µ,σρ (OT12)η,αβ .
(A.34)
What we want to achieve is that also the right-hand-side is written in terms of the full amplitudeT23. We know from Eq. (A.32) that
T23 =1
dof(η)
∑
η,λ
1√cT12cT23
(O†T23
)λ,νµ
(T23)µν
αβ(OT12)η,αβ . (A.35)
Since the projection operators are unitary, i.e.(O†T)λ,νµ
(OT )λ,µν = cT dof(λ) , (A.36)
we can write Eq. (A.34) as
T13 ∼ y41
dof(η)
∑
η,λ
1√cT12cT23
(O†T23
)λ,νµ
(T23)µν
αβ(OT12)η,αβ = y4 T23 , (A.37)
with y4 defined as
y4 :=1√
cT13cT23
1
dof(λ)
(O†T13
)λ,σγ
(O†13
)γ,ρν
(O23)µ,σρ (OT23)λ,µν . (A.38)
In the same way all other xi, yi, zi, xi, yi and zi parameters can be determined. Their definingequations are given in Eqs. (A.40 - A.81) where we have assumed that all operators OT arenormalized, i.e. cT = 1, and furthermore changed the notation of indices by removing theunderline. In section A.4 we have given some overview tables (Tab. A.2 to Tab. A.7) for thenormalized projection operators in which the indices are named in the same way as they areneeded to calculate the parameters. Note, however, that independently of the tables it holds
(Oij)µ,ρσ =
(Oij)µ,ρσ , if Pi 6= Pj
(Oij)µ,σρ , if Pi = Pj. (A.39)
because if Pi = Pj there is just one possibility of assigning the indices ρ and σ to the two identicalparticles Pi (in contrast to the general case Pi 6= Pj where – depending on the considered diagram
– in a vertex either A†i or A†j is allocated with the index ρ so that according to Eq. (3.4) one haseither a projector (Oij)µ,ρσ or a projector (Oij)µ,σρ in the expression).
x(′)1 =
1
dof(η)∑
η,λ
(O†T
(′)12
)λ,σγ
(O12
)α,σρ
(O(′)†
12
)γ,ρβ
(OT12
)η,αβ
(A.40)
115
y(′)1 =
1
dof(η)∑
η,λ
(O†T
(′)13
)λ,σγ
(O12
)α,σρ
(O(′)†
13
)γ,βρ
(OT12
)η,αβ
(A.41)
z(′)1 =
1
dof(η)∑
η,λ
(O†T
(′)23
)λ,σγ
(O12
)α,σρ
(O(′)†
23
)γ,βρ
(OT12
)η,αβ
(A.42)
x(′)2 =
1
dof(λ)(O†T
(′)12
)λ,σγ
(O12
)µ,σρ
(O(′)†
12
)γ,ρν
(OT12
)λ,µν
(A.43)
y(′)2 =
1
dof(λ)(O†T
(′)13
)λ,σγ
(O12
)µ,σρ
(O(′)†
13
)γ,νρ
(OT12
)λ,µν
(A.44)
z(′)2 =
1
dof(λ)(O†T
(′)23
)λ,σγ
(O12
)µ,σρ
(O(′)†
23
)γ,νρ
(OT12
)λ,µν
(A.45)
x(′)3 =
1
dof(λ)(O†T
(′)12
)λ,σγ
(O13
)µ,σρ
(O(′)†
12
)γ,νρ
(OT13
)λ,µν
(A.46)
y(′)3 =
1
dof(λ)(O†T
(′)13
)λ,σγ
(O13
)µ,σρ
(O(′)†
13
)γ,ρν
(OT13
)λ,µν
(A.47)
z(′)3 =
1
dof(λ)(O†T
(′)23
)λ,σγ
(O13
)µ,σρ
(O(′)†
23
)γ,ρν
(OT13
)λ,µν
(A.48)
x(′)4 =
1
dof(λ)(O†T
(′)12
)λ,σγ
(O23
)µ,σρ
(O(′)†
12
)γ,ρν
(OT23
)λ,µν
(A.49)
y(′)4 =
1
dof(λ)(O†T
(′)13
)λ,σγ
(O23
)µ,σρ
(O(′)†
13
)γ,ρν
(OT23
)λ,µν
(A.50)
z(′)4 =
1
dof(λ)(O†T
(′)23
)λ,σγ
(O23
)µ,σρ
(O(′)†
23
)γ,ρν
(OT23
)λ,µν
(A.51)
x(′)5 =
1
dof(λ)(O†T
(′)12
)λ,σγ
(O′12
)µ,σρ
(O(′)†
12
)γ,ρν
(OT ′12
)λ,µν
(A.52)
y(′)5 =
1
dof(λ)(O†T
(′)13
)λ,σγ
(O′12
)µ,σρ
(O(′)†
13
)γ,νρ
(OT ′12
)λ,µν
(A.53)
z(′)5 =
1
dof(λ)(O†T
(′)23
)λ,σγ
(O′12
)µ,σρ
(O(′)†
23
)γ,νρ
(OT ′12
)λ,µν
(A.54)
x(′)6 =
1
dof(λ)(O†T
(′)12
)λ,σγ
(O′13
)µ,σρ
(O(′)†
12
)γ,νρ
(OT ′13
)λ,µν
(A.55)
116
y(′)6 =
1
dof(λ)(O†T
(′)13
)λ,σγ
(O′13
)µ,σρ
(O(′)†
13
)γ,ρν
(OT ′13
)λ,µν
(A.56)
z(′)6 =
1
dof(λ)(O†T
(′)23
)λ,σγ
(O′13
)µ,σρ
(O(′)†
23
)γ,ρν
(OT ′13
)λ,µν
(A.57)
x(′)7 =
1
dof(λ)(O†T
(′)12
)λ,σγ
(O′23
)µ,σρ
(O(′)†
12
)γ,ρν
(OT ′23
)λ,µν
(A.58)
y(′)7 =
1
dof(λ)(O†T
(′)13
)λ,σγ
(O′23
)µ,σρ
(O(′)†
13
)γ,ρν
(OT ′23
)λ,µν
(A.59)
z(′)7 =
1
dof(λ)(O†T
(′)23
)λ,σγ
(O′23
)µ,σρ
(O(′)†
23
)γ,ρν
(OT ′23
)λ,µν
(A.60)
x(′)1 =
1
dof(η)∑
η,λ
(O†T
(′)12
)λ,σγ
(O12
)α,ρσ
(O(′)†
12
)γ,βρ
(OT12
)η,αβ
(A.61)
y(′)1 =
1
dof(η)∑
η,λ
(O†T
(′)13
)λ,σγ
(O12
)α,ρσ
(O(′)†
13
)γ,βρ
(OT12
)η,αβ
(A.62)
z(′)1 =
1
dof(η)∑
η,λ
(O†T
(′)23
)λ,σγ
(O12
)α,ρσ
(O(′)†
23
)γ,βρ
(OT12
)η,αβ
(A.63)
x(′)2 =
1
dof(λ)(O†T
(′)12
)λ,σγ
(O12
)µ,ρσ
(O(′)†
12
)γ,νρ
(OT12
)λ,µν
(A.64)
y(′)2 =
1
dof(λ)(O†T
(′)13
)λ,σγ
(O12
)µ,ρσ
(O(′)†
13
)γ,νρ
(OT12
)λ,µν
(A.65)
z(′)2 =
1
dof(λ)(O†T
(′)23
)λ,σγ
(O12
)µ,ρσ
(O(′)†
23
)γ,νρ
(OT12
)λ,µν
(A.66)
x(′)3 =
1
dof(λ)(O†T
(′)12
)λ,σγ
(O13
)µ,ρσ
(O(′)†
12
)γ,νρ
(OT13
)λ,µν
(A.67)
y(′)3 =
1
dof(λ)(O†T
(′)13
)λ,σγ
(O13
)µ,ρσ
(O(′)†
13
)γ,νρ
(OT13
)λ,µν
(A.68)
z(′)3 =
1
dof(λ)(O†T
(′)23
)λ,σγ
(O13
)µ,ρσ
(O(′)†
23
)γ,ρν
(OT13
)λ,µν
(A.69)
x(′)4 =
1
dof(λ)(O†T
(′)12
)λ,σγ
(O23
)µ,ρσ
(O(′)†
12
)γ,ρν
(OT23
)λ,µν
(A.70)
117
y(′)4 =
1
dof(λ)(O†T
(′)13
)λ,σγ
(O23
)µ,ρσ
(O(′)†
13
)γ,ρν
(OT23
)λ,µν
(A.71)
z(′)4 =
1
dof(λ)(O†T
(′)23
)λ,σγ
(O23
)µ,ρσ
(O(′)†
23
)γ,νρ
(OT23
)λ,µν
(A.72)
x(′)5 =
1
dof(λ)(O†T
(′)12
)λ,σγ
(O′12
)µ,ρσ
(O(′)†
12
)γ,νρ
(OT ′12
)λ,µν
(A.73)
y(′)5 =
1
dof(λ)(O†T
(′)13
)λ,σγ
(O′12
)µ,ρσ
(O(′)†
13
)γ,νρ
(OT ′12
)λ,µν
(A.74)
z(′)5 =
1
dof(ρ)(O†T
(′)23
)λ,σγ
(O′12
)µ,σσ
(O(′)†
23
)γ,νρ
(OT ′12
)λ,µν
(A.75)
x(′)6 =
1
dof(λ)(O†T
(′)12
)λ,σγ
(O′13
)µ,ρσ
(O(′)†
12
)γ,νρ
(OT ′13
)λ,µν
(A.76)
y(′)6 =
1
dof(λ)(O†T
(′)13
)λ,σγ
(O′13
)µ,ρσ
(O(′)†
13
)γ,νρ
(OT ′13
)λ,µν
(A.77)
z(′)6 =
1
dof(λ)(O†T
(′)23
)λ,σγ
(O′13
)µ,ρσ
(O(′)†
23
)γ,ρν
(OT ′13
)λ,µν
(A.78)
x(′)7 =
1
dof(λ)(O†T
(′)12
)λ,σγ
(O′23
)µ,ρσ
(O(′)†
12
)γ,ρν
(OT ′23
)λ,µν
(A.79)
y(′)7 =
1
dof(λ)(O†T
(′)13
)λ,σγ
(O′23
)µ,ρσ
(O(′)†
13
)γ,ρν
(OT ′23
)λ,µν
(A.80)
z(′)7 =
1
dof(λ)(O†T
(′)23
)λ,σγ
(O′23
)µ,ρσ
(O(′)†
23
)γ,νρ
(OT ′23
)λ,µν
(A.81)
A.3 x, y, z parameters and the 6-J-symbol
Wigner introduced the 3-J symbol [169],(j1 j2 Jm1 m2 M
),
to describe the coupling of two angular momenta j1 and j2 to a total angular momentum Jfor given associated magnetic quantum numbers m1, m2 and M . It is related to the usualClebsch-Gordan coefficients through [170]
(〈j1,m1| ⊗ 〈j2,m2|
)|J,M〉 = (−1)j1−j2+M
√2J + 1
(j1 j2 Jm1 m2 −M
), (A.82)
and only non-zero if the following properties are fulfilled [170]:
118
• m1 ∈ −|j1|, ..., |j1|, m2 ∈ −|j2|, ..., |j2| and M ∈ −|J |, ..., |J |,
• m1 +m2 = M ,
• |j1 − j2| ≤ J ≤ j1 + j2,
• j1 + j2 + J ∈ N.
As it is discussed in Ref. [170] there are – besides a large number of Regge symmetries – threeimportant symmetries of the 3-J symbol: it is invariant under even permutations of its columns,i.e.
(j1 j2 Jm1 m2 M
)=
(j2 J j1m2 M m1
)=
(J j1 j1
M m1 m2
), (A.83)
but receive a phase factor for odd permutations:
(j1 j2 Jm1 m2 M
)= (−1)j1+j2+J
(j2 j1 Jm2 m1 M
)= (−1)j1+j2+J
(j1 J j2m1 M m2
). (A.84)
The third property is that also changing the sign of all magnetic quantum numbers yields aphase:
(j1 j2 J−m1 −m2 −M
)= (−1)j1+j2+J
(j1 j2 Jm1 m2 M
). (A.85)
If we consider Eq. (A.8) and again use that in the Cartesian basis (~εα)α
= δαα holds (cf.Eq. (A.13)) we conclude that C can be written as
C :=
(~ε †β
)
β
(Oij)α,βγ(~ε †γ
)γ
(~εα)α
= (Oij)α,βγ . (A.86)
As it was done in the defining equations for the x, y, z parameters (Eqs. (A.40 - A.81)) we changethe index notation by removing the underline and conclude that
C = (Oij)α,βγ =(〈ji,mi| ⊗ 〈jj,mj|
)|Jij,Mij〉 , (A.87)
since phenomenology tells us that C is equal to the corresponding Clebsch-Gordan coefficient forcoupling spin ji of particle Ai and spin jj of Aj to the spin Jij of the dimer dij. Consequently,one can use Eq. (A.82) and write
(Oij)α,βγ = (−1)ji−jj+Mij√
2Jij + 1
(ji jj Jijmi mj −Mij
). (A.88)
However, in the coupled integral equations appear at the same time projectors (Oij)α,γβ withinterchanged second and third index (note, that ”interchanged“ does not mean ”renamed“ sincein each scattering process the indices are fixed by the particle allocation). Thus, the questionarises how the relation Eq. (A.88) is changed? The operator Oij couples the fields A†i and A†j
119
and according to the order of appearance of these particles within the dimer wave function (fixedin Eq. (3.4)), the second index corresponds to particle Pi and the third to Pj (cf. Lagrangiandensity in Eq. (3.23)). However, this is only true for distinguishable particles Pi and Pj. Ifthey are identical the swap of them obviously does not change the Clebsch-Gordan coefficient.Therefore the interchange of second and third index of Oij yields the relation
(Oij)α,γβ =
(〈jj,mj| ⊗ 〈ji,mi|
)|Jij,Mij〉 , for Pi 6= Pj(
〈ji,mi| ⊗ 〈jj,mj|)|Jij,Mij〉 , for Pi = Pj
, (A.89)
where the total spin Jij is of course not changed. However, in the upper case this is not ourconvention regarding the order of the constituents within a dimer wave function. Hence, weremember the relation in Eq. (A.29) and write
(Oij)α,γβ = (−1)(1−δPiPj )(Jij−ji−jj)(〈ji,mi| ⊗ 〈jj,mj|
)|Jij,Mij〉 , (A.90)
which holds for the same set of indices fixed in Eq. (A.87) and where the factor δPiPj ensuresthat nothing is changed if the particles are identical. Since the right-hand-side yields a (real)Clebsch-Gordan coefficient it is Hermitian and thus the left-hand-side must be. Consequently,Hermitian conjugation does not change Eq. (A.87). In summary we thus find with the help of
Eq. (A.88) for a (due to the particle allocation) fixed set of indices α, β, γ and i < j ∈ 1, 2, 3:
(Oij)α,βγ =(〈ji,mi| ⊗ 〈jj,mj|
)|Jij,Mij〉 = (−1)ji−jj+Mij
√2Jij + 1
(ji jj Jijmi mj −Mij
),
(A.91)
(Oij)α,γβ =(〈jj,mj| ⊗ 〈ji,mi|
)|Jij,Mij〉 = (−1)(1−δPiPj )(Jij−ji−jj)
(〈ji,mi| ⊗ 〈jj,mj|
)|Jij,Mij〉
= (−1)(1−δPiPj )(Jij−ji−jj)(−1)ji−jj+Mij√
2Jij + 1
(ji jj Jijmi mj −Mij
)
= (−1)ji−jj+Mij+(1−δPiPj )(Jij−ji−jj)√2Jij + 1
(ji jj Jijmi mj −Mij
), (A.92)
(O†ij)α,γβ
=[(Oij)α,βγ
]†=
[(−1)ji−jj+Mij
√2Jij + 1
(ji jj Jijmi mj −Mij
)]†
= (−1)ji−jj+Mij√
2Jij + 1
(ji jj Jijmi mj −Mij
), (A.93)
(O†ij)α,βγ
=[(Oij)α,γβ
]†=
[(−1)Jij−2jj+Mij
√2Jij + 1
(ji jj Jijmi mj −Mij
)]†
= (−1)ji−jj+Mij+(1−δPiPj )(Jij−ji−jj)√2Jij + 1
(ji jj Jijmi mj −Mij
). (A.94)
These results allow us to write the x, y, z parameters in terms of 3-J symbols. Before we do sokeep in mind that in all Feynman diagrams in appendix C the exchanged particle has the indexρ, i.e. ρ in the detailed notation of the current section. Hence, depending on the particle Pi, ρ
120
is either the second or the third index of the projection operator O. With our convention fixedin Eq. (3.4) it is then also fixed that
(Oij) . , . ρ =(〈ji,mi| ⊗ 〈jj,mj|
)|Jij,Mij〉
= (−1)ji−jj+Mij√
2Jij + 1
(ji jj Jijmi mj −Mij
), for i < j ∈ 1, 2, 3 ,
and the relation for (Oij) . ,ρ . is determined by Eq. (A.92). Note, that there is no conventionregarding the channel projectors OTij ; they are simply chosen in the way that the dimer spincomes first. Thus, one can apply Eq. (A.91) in the form
(OTij
)λ,µν
=(〈Jij,Mij| ⊗ 〈jk,mk|
) ∣∣J(ij)k,M(ij)k
⟩
= (−1)Jij−jk+M√
2J + 1
(Jij jk JMij mk −M
), for i < j ∈ 1, 2, 3 .
As an example for this method we again consider y4 derived in the previous section:
y4 =1
dof(λ)
∑
λ,σ,γ,µ,ν,ρ
(O†T13
)λ,σγ
(O†13
)γ,ρν
(O23)µ,σρ (OT23)λ,µν .
For simplicity we below assume that all particles are distinguishable. Nevertheless, everythingworks out in the same way if some particles are identical; one only has to keep in mind thatthe indices of some operators O might be interchanged to ensure the right order according toEq. (3.4). In terms of 3-J symbols y4 is given by
y4 =1
dof(λ)
∑
λ,σ,γ,µ,ν,ρ
(−1)J13−j2+M√
2J + 1
(J13 j2 J
M13(γ) m2(σ) −M(λ)
)
× (−1)j1−j3+M13√
2J13 + 1
(j1 j3 J13
m1(ν) m3(ρ) −M13(γ)
)
× (−1)j2−j3+M23√
2J23 + 1
(j2 j3 J23
m2(σ) m3(ρ) −M23(µ)
)
× (−1)J23−j1+M√
2J + 1
(J23 j1 J
M23(µ) m1(ν) −M(λ)
), (A.95)
where we have made the sum over spin indices explicit and used the variables J and M forthe spin channel λ and its magnetic quantum number. Since each magnetic quantum numberimplicitly depends on the corresponding index we have written m ≡ m(.) to emphasize thisfact. In this sense the summation over all spin indices can be expressed as a summation over allmagnetic quantum numbers. Furthermore, we know from quantum mechanics that (2J + 1) is
the dimension of the associated spin space and thus (2J + 1) = dof(λ) holds. Thus, Eq. (A.95)
121
simplifies to
y4 =√
(2J13 + 1)(2J23 + 1)
|j1|∑
m1=−|j1|
|j2|∑
m2=−|j2|
|j3|∑
m3=−|j3|
|J13|∑
M13=−|J13|
|J23|∑
M23=−|J23|
|J |∑
M=−|J |
× (−1)J13+J23−2j3+M13+M23+2M
(J13 j2 JM13 m2 −M
)(j2 j3 J23
m2 m3 −M23
)
×(j1 j3 J13
m1 m3 −M13
)(J23 j1 JM23 m1 −M
). (A.96)
One observes that the substitution M13 → −M13 does not change the sum in y4 since M13 ∈−|J13|, ..., |J13|:
y4 =√
(2J13 + 1)(2J23 + 1)
|j1|∑
m1=−|j1|
|j2|∑
m2=−|j2|
|j3|∑
m3=−|j3|
|J13|∑
M13=−|J13|
|J23|∑
M23=−|J23|
|J |∑
M=−|J |
× (−1)J13+J23−2j3−M13+M23+2M
(J13 j2 J−M13 m2 −M
)(j2 j3 J23
m2 m3 −M23
)
×(j1 j3 J13
m1 m3 M13
)(J23 j1 JM23 m1 −M
). (A.97)
Using the symmetry properties of the 3-J symbol given in Eq. (A.83) and Eq. (A.85) yields
y4 =√
(2J13 + 1)(2J23 + 1)
|j|∑
m=−|j|(−1)J13+J23−2j3−M13+M23+2M
× (−1)j2+J+J13
(j2 J J13
−m2 M M13
)(j2 j3 J23
m2 m3 −M23
)
× (−1)j1+j3+J13
(j1 j3 J13
−m1 −m3 −M13
)(j1 J J23
m1 −M M23
), (A.98)
with∑
m being a placeholder for all sums over magnetic quantum numbers.
Now consider the so-called 6-J symbol,
j1 j2 J12
j3 J J23
,
introduced by Wigner in the same work of Ref. [169]. It is used to describe the coupling ofthree spins j1, j2 and j3 to a total spin J with two intermediate spins J12 and J23 comingfrom the coupling of j1 with j2 and j2 with j3, respectively. More precisely, it is related to theClebsch-Gordan coefficient
〈(j1, (j2, j3)J23) J | ((j1, j2)J12, j3) J〉 , (A.99)
122
which couples the incoming spins according to j1⊗ j2 → J12 and J12⊗ j3 → J while in the ”out“state one has j2 ⊗ j3 → J23 and j1 ⊗ J23 → J . Keep in mind that the order of the respectivespins is important since due to Eq. (A.29) swapping leads to possible phase factors. FollowingRef. [170] we know that the coefficient Eq. (A.99) is related to the Racah W-coefficient via
W (j1j2Jj3; J12J23) =1√
(2J12 + 1)(2J23 + 1)〈(j1, (j2, j3)J23) J | ((j1, j2)J12, j3) J〉 , (A.100)
which itself can be written in terms of the 6-J symbol
W (j1j2Jj3; J12J23) = (−1)j1+j2+j3+J
j1 j2 J12
j3 J J23
. (A.101)
Consequently, one finds that the 6-J symbol above is proportional to the Clebsch-Gordan co-efficient for the coupling of incoming spins j1 ⊗ j2 → J12, J12 ⊗ j3 → J and outgoing spinsj2 ⊗ j3 → J23, j1 ⊗ J23 → J :
〈(j1, (j2, j3)J23) J | ((j1, j2)J12, j3) J〉 = (−1)j1+j2+j3+J√
(2J12 + 1)(2J23 + 1)
j1 j2 J12
j3 J J23
.
(A.102)
In Ref. [170] it is shown that the 6-J symbol has certain symmetries: firstly, it is invariant underany permutation of columns,j1 j2 j3
j4 j5 j6
=
j2 j3 j1
j5 j6 j4
=
j3 j1 j2
j6 j4 j5
=
j2 j1 j3
j5 j4 j6
=
j1 j3 j2
j4 j6 j5
=
j3 j2 j1
j6 j5 j4
,
(A.103)
and secondly, it is also invariant if one interchanges the upper and lower argument in each of anytwo columns [170]:
j1 j2 j3
j4 j5 j6
=
j1 j5 j6
j4 j2 j3
=
j4 j5 j3
j1 j2 j6
=
j4 j2 j6
j1 j5 j3
. (A.104)
From the considerations above it is clear that the x, y, z parameters should be related to the 6-Jsymbol. Using again our example parameter y4 one can indeed show that this is the case: fromthe diagram in Fig. A.2 to which y4 is proportional, one observes that y4 is the coefficient for thecoupling of spins (j2 ⊗ j3 → J23)⊗ j1 → J in the ”in“ state and (j1 ⊗ j3 → J13)⊗ j2 → J in theoutgoing one (remember our convention that the dimer is coupled with the left-over particle to atotal spin J and not vice versa). Comparing this with the relation in Eq. (A.102) one concludesthat one has to interchange j1 with j3 as well as j2 with J13. Thus, with the help of Eq. (A.29)one finds
y4 = 〈((j1, j3)J13, j2) J | ((j2, j3)J23, j1) J〉= (−1)J−J13−j2(−1)J13−j1−j3 〈(j2, (j3, j1)J13) J | ((j2, j3)J23, j1) J〉
= (−1)J−J13−j2(−1)J13−j1−j3(−1)j1+j2+j3+J√
(2J23 + 1)(2J13 + 1)
j2 j3 J23
j1 J J13
= (−1)2J√
(2J23 + 1)(2J13 + 1)
j1 j3 J13
j2 J J23
, (A.105)
123
d13
A3
A2
T23
d23
A1
d12
P3
Figure A.2: Diagram to which y4 is proportional.
where we have used Eq. (A.102) in the third and the symmetry properties in Eq. (A.104) in thefourth step.In order to check that Eq. (A.105) is a correct representation of the parameter y4 we will verifythat it is equivalent to what we have found in Eq. (A.97). In Ref. [170] it is shown that the 6-Jsymbol can be defined in terms of 3-J symbols:
j1 j3 J13
j2 J J23
=
|j|∑
m=−|j|(−1)j1−m1+j3−m3+J13−M13+j2−m2+J−M+J23−M23
(j1 j3 J13
−m1 −m3 −M13
)
×(j1 J J23
m1 −M M23
)(j2 j3 J23
m2 m3 −M23
)(j2 J J13
−m2 M M13
). (A.106)
Hence, we can replace the product of four 3-J symbols in Eq. (A.97) by the 6-J symbol timesthe corresponding phase factor in Eq. (A.106) to obtain
y4 =√
(2J13 + 1)(2J23 + 1)
j1 j3 J13
j2 J J23
|j|∑
m=−|j|(−1)J13+J23−2j3−M13+M23+2M
× (−1)j2+J+J13(−1)j1+j3+J13(−1)j1−m1+j3−m3+J13−M13+j2−m2+J−M+J23−M23 . (A.107)
In order to simplify the sum over the phase factor
|j|∑
m=−|j|(−1)4J13+2J23+2J+2j1+2j2−2M13+M−m1−m2−m3 ,
we use that the second condition for a non-vanishing 3-J symbol given on page 118 yields in ourcase
−m1 −m3 = M13 ⇒ −(m1 +m3) = M13
−m2 +M = −M13 ⇒ M −m2 = −M13 . (A.108)
Furthermore, the first condition on page 118 implies that among others (J13−M13) ∈ N so thatthe factor (−1)2(J13−M13) = 1. Similarly, (−1)2(J13+j2+J) = (−1)2(J23+j1+J) = 1 since the fourthcondition on page 118 tells us that the respective combinations in the exponent are also integers.
124
The phase factor is thus equal to
|j|∑
m=−|j|(−1)4J13+2J23+2J+2j1+2j2−2M13+M−m1−m2−m3 =
|j|∑
m=−|j|(−1)2J13+2J23+2J+2j1+2j2(−1)2(J13−M13)
= (−1)2J13+2J23+2J+2j1+2j2
= (−1)2(J13+j2+J)(−1)2(J23+j1+J)(−1)−2J
= (−1)−2J = (−1)2J . (A.109)
Therefore the parameter y4 in Eq. (A.107) gets the compact form
y4 = (−1)2J√
(2J13 + 1)(2J23 + 1)
j1 j3 J13
j2 J J23
, (A.110)
which is equivalent to Eq. (A.105). Hence, we conclude that indeed both representations Eq. (A.97)and Eq. (A.105) for the parameter y4 are correct.The advantage of the 6-J symbol notation in y4 is that in contrast to the definition of y4 in theprevious section (Eq. (A.38)) it is not necessary to have the projection operators O explicitlygiven. It is sufficient to know the spin of the three particles and corresponding dimers as wellas the scattering channel. This allows us to analyze systems consisting of particles with spinshigher than 1 which undergo scattering processes in channels higher than 2, where the projectionoperators are more and more complicated or even not known.In the same way as we derived Eq. (A.105) (or equivalently Eq. (A.110)) one can deduce ex-pressions for all other x, y, z parameters in terms of 6-J symbols. For instance, one can – usingEq. (A.102) – deduce x
(′)2 from
x(′)2 = δ
(12)P1P3
⟨((j2, j3)J
(′)23 , j1
)J∣∣∣ ((j1, j2)J12, j3)
⟩=⟨(
(j1, j2)J(′)12 , j1
)J∣∣∣ ((j1, j2)J12, j1)
⟩
= (−1)2J
√(2J
(′)12 + 1
)(2J12 + 1)
j1 j2 J
(′)12
j1 J J12
,
and y(′)2 from
y(′)2 = δP1P2
⟨((j2, j3)J
(′)23 , j1
)J∣∣∣ ((j1, j2)J12, j3)
⟩=⟨(
(j1, j3)J(′)13 , j1
)J∣∣∣ ((j1, j1)J12, j3)
⟩
= (−1)(1−δP1P3 )(J(′)13−j1−j3)
⟨((j3, j1)J
(′)13 , j1
)J∣∣∣ ((j1, j1)J12, j3)
⟩
= (−1)2J+(1−δP1P3)
(J(′)13−j1−j3
)√(2J
(′)13 + 1
)(2J12 + 1)
j3 j1 J
(′)13
j1 J J12
.
Note, that the Kronecker-delta in front of the Clebsch-Gordan coefficient originally stands infront of the corresponding diagram. Therefore the diagram itself is changed before one appliesthe Feynman rules and thus one must not interchange the spins within the Clebsch-Gordancoefficient in the first line; they are interchanged due to the different vertex factor one has toapply for the changed diagram. However, in the second line of y
(′)2 we indeed swap the spins in
125
the coefficient and thus pick up a phase factor (as long as P1 6= P3).The results are summarized in Eqs. (A.111 - A.146) where we notice that the parameters withindex ”1“ are identical to those with index ”2“. Keep in mind that we have derived all relationsin terms of spin only. Consequently, one has to multiply the results of the x, y, z, parameters forspin with the results for isospin to get the full spin and isospin dependent parameter. However,we do not give the isospin formulae explicitly since spin and isospin behave analogously. Hence,one simply has to replace all j’s and J ’s by i’s and I’s for the latter. As a practical remark inthe application of the 6-J symbol method, note that in Mathematica the command SixJSymbol
provides an automated calculation of 6-J symbols for given spins.
x(′)1,2 = (−1)2J
√(2J
(′)12 + 1
)(2J12 + 1)
j1 j2 J
(′)12
j1 J J12
(A.111)
y(′)1,2 = (−1)
2J+(1−δP1P3)(J(′)13−j1−j3
)√(2J
(′)13 + 1
)(2J12 + 1)
j3 j1 J
(′)13
j1 J J12
(A.112)
z(′)1,2 = (−1)
2J+(1−δP2P3)(J(′)23−j2−j3
)√(2J
(′)23 + 1
)(2J12 + 1)
j3 j2 J
(′)23
j1 J J12
(A.113)
x(′)3 = (−1)
2J+(1−δP1P2)(J(′)12−j1−j2
)√(2J
(′)12 + 1
)(2J13 + 1)
j2 j1 J
(′)12
j1 J J13
(A.114)
y(′)3 = (−1)2J
√(2J
(′)13 + 1
)(2J13 + 1)
j1 j3 J
(′)13
j1 J J13
(A.115)
z(′)3 = (−1)2J
√(2J
(′)23 + 1
)(2J13 + 1)
j2 j3 J
(′)23
j1 J J13
(A.116)
x(′)4 = (−1)2J
√(2J
(′)12 + 1
)(2J23 + 1)
j1 j2 J
(′)12
j2 J J23
(A.117)
y(′)4 = (−1)2J
√(2J
(′)13 + 1
)(2J23 + 1)
j1 j3 J
(′)13
j2 J J23
(A.118)
z(′)4 = (−1)2J
√(2J
(′)23 + 1
)(2J23 + 1)
j2 j3 J
(′)23
j2 J J23
(A.119)
x(′)5 = (−1)2J
√(2J
(′)12 + 1
)(2J ′12 + 1)
j1 j2 J
(′)12
j1 J J ′12
(A.120)
y(′)5 = (−1)
2J+(1−δP1P3)(J(′)13−j1−j3
)√(2J
(′)13 + 1
)(2J ′12 + 1)
j3 j1 J
(′)13
j1 J J ′12
(A.121)
z(′)5 = (−1)
2J+(1−δP2P3)(J(′)23−j2−j3
)√(2J
(′)23 + 1
)(2J ′12 + 1)
j3 j2 J
(′)23
j1 J J ′12
(A.122)
126
x(′)6 = (−1)
2J+(1−δP1P2)(J(′)12−j1−j2
)√(2J
(′)12 + 1
)(2J ′13 + 1)
j2 j1 J
(′)12
j1 J J ′13
(A.123)
y(′)6 = (−1)2J
√(2J
(′)13 + 1
)(2J ′13 + 1)
j1 j3 J
(′)13
j1 J J ′13
(A.124)
z(′)6 = (−1)2J
√(2J
(′)23 + 1
)(2J ′13 + 1)
j2 j3 J
(′)23
j1 J J ′13
(A.125)
x(′)7 = (−1)2J
√(2J
(′)12 + 1
)(2J ′23 + 1)
j1 j2 J
(′)12
j2 J J ′23
(A.126)
y(′)7 = (−1)2J
√(2J
(′)13 + 1
)(2J ′23 + 1)
j1 j3 J
(′)13
j2 J J ′23
(A.127)
z(′)7 = (−1)2J
√(2J
(′)23 + 1
)(2J ′23 + 1)
j2 j3 J
(′)23
j2 J J ′23
(A.128)
x(′)1,2 = (−1)
2J+(1−δP1P2)(J12+J
(′)12−2j1−2j2
)√(2J
(′)12 + 1
)(2J12 + 1)
j2 j1 J
(′)12
j2 J J12
(A.129)
y(′)1,2 = (−1)
2J+(1−δP1P2)(J12−j1−j2)+(1−δP1P3)(J(′)13−j1−j3
)√(2J
(′)13 + 1
)(2J12 + 1)
j3 j1 J
(′)13
j2 J J12
(A.130)
z(′)1,2 = (−1)
2J+(1−δP2P3)(J(′)23−j2−j3
)√(2J
(′)23 + 1
)(2J12 + 1)
j3 j2 J
(′)23
j2 J J12
(A.131)
x(′)3 = (−1)
2J+(1−δP1P3)(J13−j1−j3)+(1−δP1P2)(J(′)12−j1−j2
)√(2J
(′)12 + 1
)(2J13 + 1)
j2 j1 J
(′)12
j3 J J13
(A.132)
y(′)3 = (−1)
2J+(1−δP1P3)(J13+J
(′)13−2j1−2j3
)√(2J
(′)13 + 1
)(2J13 + 1)
j3 j1 J
(′)13
j3 J J13
(A.133)
z(′)3 = (−1)2J
√(2J
(′)23 + 1
)(2J13 + 1)
j2 j3 J
(′)23
j3 J J13
(A.134)
x(′)4 = (−1)2J+(1−δP2P3)(J23−j2−j3)
√(2J
(′)12 + 1
)(2J23 + 1)
j1 j2 J
(′)12
j3 J J23
(A.135)
y(′)4 = (−1)2J
√(2J
(′)13 + 1
)(2J23 + 1)
j1 j3 J
(′)13
j3 J J23
(A.136)
z(′)4 = (−1)
2J+(1−δP2P3)(J23+J
(′)23−2j2−2j3
)√(2J
(′)23 + 1
)(2J23 + 1)
j3 j2 J
(′)23
j3 J J23
(A.137)
127
x(′)5 = (−1)
2J+(1−δP1P2)(J12+J
(′)12−2j1−2j2
)√(2J
(′)12 + 1
)(2J ′12 + 1)
j2 j1 J
(′)12
j2 J J ′12
(A.138)
y(′)5 = (−1)
2J+(1−δP1P2)(J ′12−j1−j2)+(1−δP1P3)(J(′)13−j1−j3
)√(2J
(′)13 + 1
)(2J ′12 + 1)
j3 j1 J
(′)13
j2 J J ′12
(A.139)
z(′)5 = (−1)
2J+(1−δP2P3)(J(′)23−j2−j3
)√(2J
(′)23 + 1
)(2J ′12 + 1)
j3 j2 J
(′)23
j2 J J ′12
(A.140)
x(′)6 = (−1)
2J+(1−δP1P3)(J ′13−j1−j3)+(1−δP1P2)(J(′)12−j1−j2
)√(2J
(′)12 + 1
)(2J ′13 + 1)
j2 j1 J
(′)12
j3 J J ′13
(A.141)
y(′)6 = (−1)
2J+(1−δP1P3)(J13+J
(′)13−2j1−2j3
)√(2J
(′)13 + 1
)(2J ′13 + 1)
j1 j3 J
(′)13
j1 J J ′13
(A.142)
z(′)6 = (−1)2J
√(2J
(′)23 + 1
)(2J ′13 + 1)
j2 j3 J
(′)23
j3 J J ′13
(A.143)
x(′)7 = (−1)2J+(1−δP2P3)(J ′23−j2−j3)
√(2J
(′)12 + 1
)(2J ′23 + 1)
j1 j2 J
(′)12
j3 J J ′23
(A.144)
y(′)7 = (−1)2J
√(2J
(′)13 + 1
)(2J ′23 + 1)
j1 j3 J
(′)13
j3 J J ′23
(A.145)
z(′)7 = (−1)
2J+(1−δP2P3)(J23+J
(′)23−2j2−2j3
)√(2J
(′)23 + 1
)(2J ′23 + 1)
j3 j2 J
(′)23
j3 J J ′23
(A.146)
128
A.4 Projection operator summary tables
J (Ai)β ⊗ J (Aj)γ → J (dij)α index assignment projection operator (Oij)α,βγ
0 ⊗ 0 → 0
α = spin 0 index
β = spin 0 indexγ = spin 0 index
1
0 ⊗ 12→ 1
2
α = α spin 1/2 index
β = spin 0 indexγ = γ spin 1/2 index
δαγ
0 ⊗ 1 → 1
α = i spin 1 index
β = spin 0 indexγ = k spin 1 index
δik
12⊗ 0 → 1
2
α = α spin 1/2 index
β = β spin 1/2 indexγ = spin 0 index
δαβ
12⊗ 1
2→ 0
α = spin 0 index
β = β spin 1/2 indexγ = γ spin 1/2 index
i√2
(σ2)βγ
12⊗ 1
2→ 1
α = i spin 1 index
β = β spin 1/2 indexγ = γ spin 1/2 index
i√2
(σiσ2)βγ
12⊗ 1 → 1
2
α = α spin 1/2 index
β = β spin 1/2 indexγ = k spin 1 index
1√3
(σk)βα
1 ⊗ 0 → 1
α = i spin 1 index
β = j spin 1 indexγ = spin 0 index
δij
1 ⊗ 12→ 1
2
α = α spin 1/2 index
β = j spin 1 indexγ = γ spin 1/2 index
− 1√3
(σj)γα
1 ⊗ 1 → 0
α = spin 0 index
β = j spin 1 indexγ = k spin 1 index
− 1√3δjk
1 ⊗ 1 → 1
α = i spin 1 index
β = j spin 1 indexγ = k spin 1 index
− 1√2
(Ui)jk
Table A.1: Summary of spin projection operators where the indices α, β, γ... ∈ 1, 2 re-present spin 1
2, indices i, j, k... ∈ 1, 2, 3 represent spin 1, σi are the SU(2) generators (Pauli
matrices) and Ui are the generators of SO(3).
129
J (Ai)σ ⊗ J (Aj)ρ → J(d
(′)ij
)µ
index assignment projection operator(O(′)ij
)µ,σρ
0 ⊗ 0 → 0µ = spin 0 indexσ = spin 0 indexρ = spin 0 index
1
0 ⊗ 12→ 1
2
µ = µ spin 1/2 indexσ = spin 0 indexρ = ρ spin 1/2 index
δµρ
0 ⊗ 1 → 1µ = m spin 1 indexσ = spin 0 indexρ = r spin 1 index
δmr
12⊗ 0 → 1
2
µ = µ spin 1/2 indexσ = σ spin 1/2 indexρ = spin 0 index
δµσ
12⊗ 1
2→ 0
µ = spin 0 indexσ = σ spin 1/2 indexρ = ρ spin 1/2 index
i√2
(σ2)σρ
12⊗ 1
2→ 1
µ = m spin 1 indexσ = σ spin 1/2 indexρ = ρ spin 1/2 index
i√2
(σmσ2)σρ
12⊗ 1 → 1
2
µ = µ spin 1/2 indexσ = σ spin 1/2 indexρ = r spin 1 index
1√3
(σr)σµ
1 ⊗ 0 → 1µ = m spin 1 indexσ = s spin 1 indexρ = spin 0 index
δms
1 ⊗ 12→ 1
2
µ = µ spin 1/2 indexσ = s spin 1 indexρ = ρ spin 1/2 index
− 1√3
(σs)ρµ
1 ⊗ 1 → 0µ = spin 0 indexσ = s spin 1 indexρ = r spin 1 index
− 1√3δsr
1 ⊗ 1 → 1µ = m spin 1 indexσ = s spin 1 indexρ = r spin 1 index
− 1√2
(Um)sr
Table A.2: Summary of spin projection operators where the indices are chosen according totheir appearance in the x, y, z parameters Eqs. (A.40 - A.81).
130
J (Ai)ρ ⊗ J (Aj)σ → J(d
(′)ij
)µ
index assignment projection operator(O(′)ij
)µ,ρσ
0 ⊗ 0 → 0µ = spin 0 indexρ = spin 0 indexσ = spin 0 index
1
0 ⊗ 12→ 1
2
µ = µ spin 1/2 indexρ = spin 0 indexσ = σ spin 1/2 index
δµσ
0 ⊗ 1 → 1µ = m spin 1 indexρ = spin 0 indexσ = s spin 1 index
δms
12⊗ 0 → 1
2
µ = µ spin 1/2 indexρ = ρ spin 1/2 indexσ = spin 0 index
δµρ
12⊗ 1
2→ 0
µ = spin 0 indexρ = ρ spin 1/2 indexσ = σ spin 1/2 index
i√2
(σ2)ρσ
12⊗ 1
2→ 1
µ = m spin 1 indexρ = ρ spin 1/2 indexσ = σ spin 1/2 index
i√2
(σmσ2)ρσ
12⊗ 1 → 1
2
µ = µ spin 1/2 indexρ = ρ spin 1/2 indexσ = s spin 1 index
1√3
(σs)ρµ
1 ⊗ 0 → 1µ = m spin 1 indexρ = r spin 1 indexσ = spin 0 index
δmr
1 ⊗ 12→ 1
2
µ = µ spin 1/2 indexρ = r spin 1 indexσ = σ spin 1/2 index
− 1√3
(σr)σµ
1 ⊗ 1 → 0µ = spin 0 indexρ = r spin 1 indexσ = s spin 1 index
− 1√3δrs
1 ⊗ 1 → 1µ = m spin 1 indexρ = r spin 1 indexσ = s spin 1 index
− 1√2
(Um)rs
Table A.3: Summary of spin projection operators where the indices are chosen according totheir appearance in the x, y, z parameters Eqs. (A.40 - A.81).
131
J (Ai)ν ⊗ J (Aj)ρ → J(d
(′)ij
)γ
index assignment projection operator(O(′)†ij
)γ,ρν
0 ⊗ 0 → 0γ = spin 0 indexν = spin 0 indexρ = spin 0 index
1
0 ⊗ 12→ 1
2
γ = γ spin 1/2 indexν = spin 0 indexρ = ρ spin 1/2 index
δργ
0 ⊗ 1 → 1γ = g spin 1 indexν = spin 0 indexρ = r spin 1 index
δrg
12⊗ 0 → 1
2
γ = γ spin 1/2 indexν = ν spin 1/2 indexρ = spin 0 index
δνγ
12⊗ 1
2→ 0
γ = spin 0 indexν = ν spin 1/2 indexρ = ρ spin 1/2 index
− i√2
(σ2)ρν
12⊗ 1
2→ 1
γ = g spin 1 indexν = ν spin 1/2 indexρ = ρ spin 1/2 index
− i√2
(σ2σg)ρν
12⊗ 1 → 1
2
γ = γ spin 1/2 indexν = ν spin 1/2 indexρ = r spin 1 index
1√3
(σr)γν
1 ⊗ 0 → 1γ = g spin 1 indexν = n spin 1 indexρ = spin 0 index
δng
1 ⊗ 12→ 1
2
γ = γ spin 1/2 indexν = n spin 1 indexρ = ρ spin 1/2 index
− 1√3
(σn)γρ
1 ⊗ 1 → 0γ = spin 0 indexν = n spin 1 indexρ = r spin 1 index
− 1√3δrn
1 ⊗ 1 → 1γ = g spin 1 indexν = n spin 1 indexρ = r spin 1 index
− 1√2
(Ug)rn
Table A.4: Summary of spin projection operators where the indices are chosen according totheir appearance in the x, y, z parameters Eqs. (A.40 - A.81).
132
J (Ai)ρ ⊗ J (Aj)ν → J(d
(′)ij
)γ
index assignment projection operator(O(′)†ij
)γ,νρ
0 ⊗ 0 → 0γ = spin 0 indexρ = spin 0 indexν = spin 0 index
1
0 ⊗ 12→ 1
2
γ = γ spin 1/2 indexρ = spin 0 indexν = ν spin 1/2 index
δνγ
0 ⊗ 1 → 1γ = g spin 1 indexρ = spin 0 indexν = n spin 1 index
δng
12⊗ 0 → 1
2
γ = γ spin 1/2 indexρ = ρ spin 1/2 indexν = spin 0 index
δργ
12⊗ 1
2→ 0
γ = spin 0 indexρ = ρ spin 1/2 indexν = ν spin 1/2 index
− i√2
(σ2)νρ
12⊗ 1
2→ 1
γ = g spin 1 indexρ = ρ spin 1/2 indexν = ν spin 1/2 index
− i√2
(σ2σg)νρ
12⊗ 1 → 1
2
γ = γ spin 1/2 indexρ = ρ spin 1/2 indexν = n spin 1 index
1√3
(σn)γρ
1 ⊗ 0 → 1γ = g spin 1 indexρ = r spin 1 indexν = spin 0 index
δrg
1 ⊗ 12→ 1
2
γ = γ spin 1/2 indexρ = r spin 1 indexν = ν spin 1/2 index
− 1√3
(σr)γν
1 ⊗ 1 → 0γ = spin 0 indexρ = r spin 1 indexν = n spin 1 index
− 1√3δnr
1 ⊗ 1 → 1γ = g spin 1 indexρ = r spin 1 indexν = n spin 1 index
− 1√2
(Ug)nr
Table A.5: Summary of spin projection operators where the indices are chosen according totheir appearance in the x, y, z parameters Eqs. (A.40 - A.81).
133
J(d
(′)ij
)µ⊗ J (Ak)ν → channel λ index assignment projection operator
(OT
(′)ij
)λ,µν
0 ⊗ 0 → 0λ = spin 0 indexµ = spin 0 indexν = spin 0 index
1
0 ⊗ 12→ 1
2
λ = λ spin 1/2 indexµ = spin 0 indexν = ν spin 1/2 index
δλν
0 ⊗ 1 → 1λ = ` spin 1 indexµ = spin 0 indexν = n spin 1 index
δ`n
12⊗ 0 → 1
2
λ = λ spin 1/2 indexµ = µ spin 1/2 indexν = spin 0 index
δλµ
12⊗ 1
2→ 0
λ = spin 0 indexµ = µ spin 1/2 indexν = ν spin 1/2 index
i√2
(σ2)µν
12⊗ 1
2→ 1
λ = ` spin 1 indexµ = µ spin 1/2 indexν = ν spin 1/2 index
i√2
(σ`σ2)µν
12⊗ 1 → 1
2
λ = λ spin 1/2 indexµ = µ spin 1/2 indexν = n spin 1 index
1√3
(σn)µλ
12⊗ 1 → 3
2
λ = `λ spin 3/2 indexµ = µ spin 1/2 indexν = n spin 1 index
13
[(σ`σn)µλ + δ`nδµλ
]
1 ⊗ 0 → 1λ = ` spin 1 indexµ = m spin 1 indexν = spin 0 index
δ`m
1 ⊗ 12→ 1
2
λ = λ spin 1/2 indexµ = m spin 1 indexν = ν spin 1/2 index
− 1√3
(σm)νλ
1 ⊗ 12→ 3
2
λ = `λ spin 3/2 indexµ = m spin 1 indexν = ν spin 1/2 index
13
[(σ`σm)νλ + δ`mδνλ]
1 ⊗ 1 → 0λ = spin 0 indexµ = m spin 1 indexν = n spin 1 index
− 1√3δmn
1 ⊗ 1 → 1λ = ` spin 1 indexµ = m spin 1 indexν = n spin 1 index
− 1√2
(U`)mn
1 ⊗ 1 → 2λ = `k spin 2 indexµ = m spin 1 indexν = n spin 1 index
12
[δ`mδkn + δ`nδkm − 2
3δ`kδmn
]
Table A.6: Summary of spin projectors where the indices are chosen according to their appearance inthe x, y, z parameters Eqs. (A.40 - A.81). 134
J(d
(′)ij
)γ⊗ J (Ak)σ → channel λ index assignment projection operator
(O†T
(′)ij
)λ,σγ
0 ⊗ 0 → 0λ = spin 0 indexγ = spin 0 indexσ = spin 0 index
1
0 ⊗ 12→ 1
2
λ = λ spin 1/2 indexγ = spin 0 indexσ = σ spin 1/2 index
δσλ
0 ⊗ 1 → 1λ = ` spin 1 indexγ = spin 0 indexσ = s spin 1 index
δs`
12⊗ 0 → 1
2
λ = λ spin 1/2 indexγ = γ spin 1/2 indexσ = spin 0 index
δγλ
12⊗ 1
2→ 0
λ = spin 0 indexγ = γ spin 1/2 indexσ = σ spin 1/2 index
− i√2
(σ2)σγ
12⊗ 1
2→ 1
λ = ` spin 1 indexγ = γ spin 1/2 indexσ = σ spin 1/2 index
− i√2
(σ2σ`)σγ
12⊗ 1 → 1
2
λ = λ spin 1/2 indexγ = γ spin 1/2 indexσ = s spin 1 index
1√3
(σs)λγ
12⊗ 1 → 3
2
λ = `λ spin 3/2 indexγ = γ spin 1/2 indexσ = s spin 1 index
13
[(σsσ`)λγ + δs`δλγ
]
1 ⊗ 0 → 1λ = ` spin 1 indexγ = g spin 1 indexσ = spin 0 index
δg`
1 ⊗ 12→ 1
2
λ = λ spin 1/2 indexγ = g spin 1 indexσ = σ spin 1/2 index
− 1√3
(σg)λσ
1 ⊗ 12→ 3
2
λ = `λ spin 3/2 indexγ = g spin 1 indexσ = σ spin 1/2 index
13
[(σgσ`)λσ + δg`δλσ
]
1 ⊗ 1 → 0λ = spin 0 indexγ = g spin 1 indexσ = s spin 1 index
− 1√3δsg
1 ⊗ 1 → 1λ = ` spin 1 indexγ = g spin 1 indexσ = s spin 1 index
− 1√2
(U`)sg
1 ⊗ 1 → 2λ = `k spin 2 indexγ = g spin 1 indexσ = s spin 1 index
12
[δg`δsk + δs`δgk − 2
3δk`δsg
]
Table A.7: Summary of spin projectors where the indices are chosen according to their appearance inthe x, y, z parameters Eqs. (A.40 - A.81). 135
Appendix B
Symmetry factors
In this section we will determine the symmetry factors of the different diagram topologies ap-pearing in this work. For this purpose we completely ignore spin and isospin degrees of freedomand consider without loss of generality only the part dij(x) ∼ Aj(x)Ai(x) of the dimer wavefunction. Namely, we ignore the terms proportional to Aj(x)Ai(x) etc. since they lead to thesame symmetry factors. The notation follows section A.1 where we have fixed in Eq. (A.1) howbra or ket vectors with arbitrary momentum are related to the vacuum state |0〉. Since the field
A(†)(x) ∼ a(†)p is related by a Fourier transform to the respective creation or annihilation opera-
tors we use the name of the fields (A(†)i , d
(†)ij , ...) as a representation for those (note, that one can
distinguish them by their argument: the field depends on coordinate variables and the creationand annihilation operators depend on momenta). The only important property of a particle inthis section is whether it is a fermion or a boson since the former can cause additional minussigns. Therefore it is useful to extend the notation and replace Ai by Fi if the correspondingparticle is a fermion. In the same way we write for fermionic dimers fij instead of dij where thelatter is still used for bosons. The equal time anticommutators of two fermion fields are thengiven by (cf. Ref. [1])
Fi(x), F †j (y)
= δij δ
(3)(x− y) ,Fi(x), Fj(y)
=F †i (x), F †j (y)
= 0 . (B.1)
Consequently, the creation and annihilation operator (also for fermions there is just one, re-spectively, since we consider a non-relativistic theory without anti-particles) obey equal timeanticommutators [1],
(Fi
)(p)
α,(F †j
)(q)
β
= δij δαβ (2π)3δ(3)(p− q) ,
(Fi
)(p)
α,(Fj
)(q)
β
=
(F †i
)(p)
α,(F †j
)(q)
β
= 0 , (B.2)
where we have added combined spin and isospin indices α and β for completeness although wewill neglect these degrees of freedom in the following. Furthermore, we have used – as explained
136
above – the same symbol F for the operators as for the fields themselves, but with explicit de-pendence on the momenta. Below we will neglect this explicit dependence and instead write inthis section Fi(x) if the field itself is meant.In simple words, the interchange of any two fermion fields or operators yields a factor of (−1).Only in vertices it holds for operator combinations like (d†ijFjFi) that for Pi = Pj the two nowidentical fermion operators Fi within this vertex commute since they are annihilated at the samepoint and time. As a remark, note that there are of course no spinless fermions in nature sothe latter statement is a simplification: more precisely one observes that two identical fermionsin such a vertex must be in different spin states due to the Pauli exclusion principle, that is,(d†ij)α(O†ij)α,γβ(Fi)γ(Fi)β. Adding a third fermion (F †k )η from the right to the system it is clear
that one can directly contract (Fi)β with (F †k )η. For the other contraction (yielding the samediagram) where γ and η has to be coupled one picks up a minus sign from interchanging (Fi)βand (Fi)γ. However, this minus sign is canceled by the projection operator Oij 6= 1 which is forparticles with spin not trivial. At the end one always finds a factor of ”2“ for contractions likethat. This factor would be absent if one particle in the vertex is Fj 6= Fi so that the particles are
distinguishable because the contractions of (F †k )η with these fields lead to two different diagrams.Hence, we conclude that the simplification mentioned above leads to the correct result and isthus justified. Furthermore, one has to keep in mind that a contracted pair of two fermions,FiF
†j , commutes with all other fields.
In the following we have to deal with three different topologies which define the symmetry factorsSij, Sel and Sijk. The procedure will be to write down the matrix element of the amplitude T foreach diagram topology with regard to identical particles. Then one can analyze all combinationsof fermionic and bosonic fields which are possible in the considered diagram so that one candetermine the right contractions and thus its symmetry factor S via T ∼ 〈out| ... |in〉 ∼ S.The LaTex stylefile simplewick.sty used to type the contractions in this work was provided inRef. [171].
B.1 Symmetry factor Sij of self-energy diagrams
The topology of self-energy diagrams is shown in Fig. B.1. Incoming and outgoing dimers dijhave the same momentum p and the particles in the loop Ai and Aj are distinguishable if Pi 6= Pj,but in the special case where Pi = Pj it holds that Aj ≡ Ai. Keep in mind that the order ofthe operators Ai and Aj is fixed by the convention dij ∼ AjAi for i < j ∈ 1, 2, 3 in Eq. (3.4).Thus, for all dimers dij with i < j the self-energy amplitude TSE can be read off from Fig. B.1:
TSE ∼ 〈p|(d†ijO†ijAjAi
)(A†iA
†jOijdij
)|p〉
= 〈0| dij(d†ijAjAi
)(A†iA
†jdij
)d†ij |0〉 , ∀ i < j ∈ 1, 2, 3 , (B.3)
where we have used in the second step that the projection operators Oij = 1 for spin- andisospinless fields. Before we continue we have to distinguish two cases, namely whether Pi isidentical to Pj or not. We start with the first.
137
p p
Ai
Aj
dij diji TSE =
Figure B.1: Topology of self-energy diagrams with equal incoming and outgoing momentum p.The particles in the loop can be either distinguishable (Ai 6= Aj) or identical (Aj ≡ Ai).
Case 1: Pi 6= Pj
The symmetry factor Sij depends on the particle species. Thus, we have to consider four addi-tional cases regarding the fermionic or bosonic nature of the particles:
• only bosons:
TSE ∼ 〈0|dij(d†ijAjAi)(A†iA†jdij)d†ij|0〉 = 〈0|dijd†ijAiA†iAjA†jdijd†ij|0〉 ∼ Sij = +1 ,
• Ai = Fi is a fermion ⇒ dij = fij is a fermion:
TSE ∼ 〈0|fij(f †ijAjFi)(F †i A†jfij)f †ij|0〉 = 〈0|fijf †ijFiF †i AjA†jfijf †ij|0〉 ∼ Sij = +1 ,
• Aj = Fj is a fermion ⇒ dij = fij is a fermion:
TSE ∼ 〈0|fij(f †ijFjAi)(A†iF †j fij)f †ij|0〉 = 〈0|fijf †ijAiA†iFjF †j fijf †ij|0〉 ∼ Sij = +1 ,
• Ai = Fi, Aj = Fj are fermions:
TSE ∼ 〈0|dij(d†ijFjFi)(F †i F †j dij)d†ij|0〉 = 〈0|dijd†ijFiF †i FjF †j dijd†ij|0〉 ∼ Sij = +1 .
Next, we consider the case with identical particles.
Case 2: Pi = Pj
Since Pi = Pj the field Aj is identical to Ai. Thus, there are only two additional cases left: eitherAi is a fermion or not. However, now there are more allowed contractions leading to the samediagram. We find:
• only bosons:
TSE ∼ 〈0|dijd†ij[(AiAi)(A†iA†i ) + (AiAi)(A†iA†i )]dijd
†ij|0〉
= 2 〈0|dijd†ijAiA†iAiA†idijd†ij|0〉 ∼ Sij = +2 ,
138
i Tel =dij
Ai
Aj
Ai
Aj
p
q
p
q
Figure B.2: Topology of a generic two-body scattering process. Since elastic scattering is con-sidered the momenta p and q are equal in initial and final state. The scattered particles can beeither distinguishable (Ai 6= Aj) or identical (Aj ≡ Ai).
• Ai = Fi is a fermion:
TSE ∼ 〈0|dijd†ij[(FiFi)(F †i F †i ) + (FiFi)(F†i F†i )]dijd
†ij|0〉
= 2 〈0|dijd†ijFiF †i FiF †i dijd†ij|0〉 ∼ Sij = +2 .
In summary we conclude that there are no minus signs independently of the number of fermionsin the system and that for identical particles one gets an additional factor of two:
Sij =
2 , if Pi = Pj
1 , if Pi 6= Pj. (B.4)
B.2 Symmetry factor Sel of two-body elastic scattering
diagrams
We consider the elastic scattering of two particles Pi and Pj. A generic diagram for this processis shown in Fig. B.2. For i < j ∈ 1, 2, 3 the corresponding amplitude Tel is proportional to
Tel ∼ 〈p,q|(A†iA
†jOijdij
)(d†ijOijAjAi
)|p,q〉
= 〈0|AjAi(A†iA
†jdij
)(d†ijAjAi
)A†iA
†j |0〉 , ∀ i < j ∈ 1, 2, 3 , (B.5)
where the projection operators are set to 1 since we ignore spin and isospin. Again, one hasto analyze the two cases with distinguishable or identical particles separately. However, inboth cases one has to untangle the contractions to find the symmetry factor Sel for all possiblediagrams.
Case 1: Pi 6= Pj
We have to deal with four combinations of bosons and/or fermions:
• only bosons:
Tel ∼ 〈0|AjAi(A†iA†jdij)(d†ijAjAi)A†iA†j |0〉 = 〈0|AjA†jAiA†idijd†ijAiA†iAjA†j |0〉 ∼ Sel = +1 ,
139
• Ai = Fi is a fermion ⇒ dij = fij is a fermion:
Tel ∼ 〈0|AjFi(F †i A†jfij)(f †ijAjFi)F †i A†j |0〉 = 〈0|AjA†jFiF †i fijf †ijFiF †i AjA†j |0〉 ∼ Sel = +1 ,
• Aj = Fj is a fermion ⇒ dij = fij is a fermion:
Tel ∼ 〈0|FjAi(A†iF †j fij)(f †ijFjAi)A†iF †j |0〉 = 〈0|FjF †jAiA†ifijf †ijAiA†iFjF †j |0〉 ∼ Sel = +1 ,
• Ai = Fi, Aj = Fj are fermions:
Tel ∼ 〈0|FjFi(F †i F †j dij)(d†ijFjFi)F †i F †j |0〉 = 〈0|FjF †j FiF †i dijd†ijFiF †i FjF †j |0〉 ∼ Sel = +1 .
And for Pi = Pj one finds the results below.
Case 2: Pi = Pj
We have to deal with just two combinations, either only bosons or only fermions. However, thereexist four different contractions:
• only bosons:
Tel ∼ 〈0|AiAi(A†iA†idij)(d†ijAiAi)A†iA†i |0〉+ 〈0|AiAi(A†iA†idij)(d†ijAiAi)A†iA†i |0〉
+ 〈0|AiAi(A†iA†idij)(d†ijAiAi)A†iA†i |0〉+ 〈0|AiAi(A†iA†idij)(d†ijAiAi)A†iA†i |0〉
= 4 〈0|AiA†iAiA†idijd†ijAiA†iAiA†i |0〉 ∼ Sel = +4 ,
• Ai = Fi is a fermion:
Tel ∼ 〈0|FiFi(F †i F †i dij)(d†ijFiFi)F †i F †i |0〉+ 〈0|FiFi(F †i F †i dij)(d†ijFiFi)F †i F †i |0〉
+ 〈0|FiFi(F †i F †i dij)(d†ijFiFi)F †i F †i |0〉+ 〈0|FiFi(F †i F †i dij)(d†ijFiFi)F †i F †i |0〉
= 4 〈0|FiF †i FiF †i dijd†ijFiF †i FiF †i |0〉 ∼ Sel = +4 .
As for the self-energy diagram also here no additional minus signs appear due to fermionicconstituents of the dimer dij. However, in contrast to the previous section identical particlesyield a factor of ”4“ and thus one finds in summary:
Sel =
4 , if Pi = Pj
1 , if Pi 6= Pj. (B.6)
140
djk
dij
Ak
Aj
Ai
i Tijk =
q
p k
r
Figure B.3: Topology of a generic exchange diagram in dimer–particle scattering with incomingmomenta p, q and outgoing ones k, r. All three fields Ai, Aj and Ak are in general consideredas distinguishable, but may be identical (e.g. Aj ≡ Ai) in some special cases. Note, that thereare no constraints for the particle indices i, j, k ∈ 1, 2, 3.
B.3 Symmetry factor Sijk of dimer–particle scattering
diagrams
In dimer–particle scattering diagrams whose topology is shown in Fig. B.3 one has to be morecarefully with the indices of the particles. In the previous two subsections it only appearedone dimer in each diagram and since we defined the dimer states dij only for i < j ∈ 1, 2, 3(cf. section 3.1) there was no doubt about the right order of the constituents. However, inthe diagram Fig. B.3 and the corresponding amplitude Tijk the hierarchy of the indices is notfixed, that is, i, j, k ∈ 1, 2, 3 and there are no constraints like i < j. Consider for examplethe diagrams with on the one hand i = 1 and j = 2 and on the other hand with i = 2 andj = 1. Both topologies are allowed and do appear in our considerations. As the order of theconstituents of the dimers d21 ≡ d12 and d12 themselves is fixed to be A2A1 (see Eq. (3.4)), weconclude that in this section the order of Ai and Aj depends on the hierarchy of their indices.For the dimer dij and the projection operator Oij (which will be set to 1 as we ignore spin andisospin anyway) the order of their indices does not matter as indicated above. However, due thealready mentioned convention in Eq. (3.4), dij ∼ AjAi for i < j ∈ 1, 2, 3, the order of Ai andAj is fixed. Hence, the operator with the larger index appears firstly. The amplitude Tijk is thusdifferent for varying hierarchy of the indices i, j, k ∈ 1, 2, 3. One finds
Tijk ∼ 〈k, r| d†jkO†jk
(AkAj) , j ≤ k(AjAk) , j > k
(A†iA
†j) , i ≤ j
(A†jA†i ) , i > j
Oijdij |p,q〉
= 〈0| djkAid†jk
(AkAj)(A†iA†j) , i ≤ j ≤ k
(AkAj)(A†jA†i ) , i > j ≤ k
(AjAk)(A†iA†j) , i ≤ j > k
(AjAk)(A†jA†i ) , i > j > k
dijd
†ijA†k |0〉 , (B.7)
141
where we have set the projection operator O to 1 and combined the conditions of the vertexcontributions. Furthermore, one can move A†k two steps to left which does not yield any minus
sign independently of the particle species and already contract the two dimer operators dijd†ij on
the right:
Tijk ∼ 〈0| djkAid†jk
(AkAj)(A†iA†j) , i ≤ j ≤ k
(AkAj)(A†jA†i ) , i > j ≤ k
(AjAk)(A†iA†j) , i ≤ j > k
(AjAk)(A†jA†i ) , i > j > k
A†kdijd
†ij |0〉 . (B.8)
One proceeds and analyzes all contractions for different combinations of fermions and bosonswith regard to possibly identical particles. We start with the case where all particles are dis-tinguishable.
Case 1: Pi 6= Pj 6= Pk
Here all operators Ai, Aj and Ak are distinguishable so one has to consider eight different boson/ fermion configurations:
• only bosons:
Tijk ∼ 〈0| djkd†jk
Ai(AkAj)(A†iA†j)A
†k , i ≤ j ≤ k
Ai(AkAj)(A†jA†i )A
†k , i > j ≤ k
Ai(AjAk)(A†iA†j)A
†k , i ≤ j > k
Ai(AjAk)(A†jA†i )A
†k , i > j > k
dijd†ij |0〉
= 〈0| djkd†jk
AiA†iAjA
†jAkA
†k , i ≤ j ≤ k
AiA†iAjA
†jAkA
†k , i > j ≤ k
AiA†iAjA
†jAkA
†k , i ≤ j > k
AiA†iAjA
†jAkA
†k , i > j > k
dijd†ij |0〉
∼ Sijk = +1 ,
142
• Ai = Fi is a fermion ⇒ dij = fij is a fermion:
Tijk ∼ 〈0| djkd†jk
Fi(AkAj)(F†i A†j)A
†k , i ≤ j ≤ k
Fi(AkAj)(A†jF†i )A†k , i > j ≤ k
Fi(AjAk)(F†i A†j)A
†k , i ≤ j > k
Fi(AjAk)(A†jF†i )A†k , i > j > k
fijf†ij |0〉
= 〈0| djkd†jk
FiF†i AjA
†jAkA
†k , i ≤ j ≤ k
FiF†i AjA
†jAkA
†k , i > j ≤ k
FiF†i AjA
†jAkA
†k , i ≤ j > k
FiF†i AjA
†jAkA
†k , i > j > k
fijf†ij |0〉
∼ Sijk = +1 ,
• Aj = Fj is a fermion ⇒ dij = fij, djk = fjk are fermions:
Tijk ∼ 〈0| fjkf †jk
Ai(AkFj)(A†iF†j )A†k , i ≤ j ≤ k
Ai(AkFj)(F†jA†i )A
†k , i > j ≤ k
Ai(FjAk)(A†iF†j )A†k , i ≤ j > k
Ai(FjAk)(F†jA†i )A
†k , i > j > k
fijf†ij |0〉
= 〈0| fjkf †jk
AiA†iFjF
†jAkA
†k , i ≤ j ≤ k
AiA†iFjF
†jAkA
†k , i > j ≤ k
AiA†iFjF
†jAkA
†k , i ≤ j > k
AiA†iFjF
†jAkA
†k , i > j > k
fijf†ij |0〉
∼ Sijk = +1 ,
143
• Ak = Fk is a fermion ⇒ djk = fjk is a fermion:
Tijk ∼ 〈0| fjkf †jk
Ai(FkAj)(A†iA†j)F
†k , i ≤ j ≤ k
Ai(FkAj)(A†jA†i )F
†k , i > j ≤ k
Ai(AjFk)(A†iA†j)F
†k , i ≤ j > k
Ai(AjFk)(A†jA†i )F
†k , i > j > k
dijd†ij |0〉
= 〈0| fjkf †jk
AiA†iAjA
†jFkF
†k , i ≤ j ≤ k
AiA†iAjA
†jFkF
†k , i > j ≤ k
AiA†iAjA
†jFkF
†k , i ≤ j > k
AiA†iAjA
†jFkF
†k , i > j > k
dijd†ij |0〉
∼ Sijk = +1 ,
• Ai = Fi, Aj = Fj are fermions ⇒ djk = fjk is a fermion:
Tijk ∼ 〈0| − fjkf †jk
Fi(AkFj)(F†i F†j )A†k , i ≤ j ≤ k
Fi(AkFj)(F†j F†i )A†k , i > j ≤ k
Fi(FjAk)(F†i F†j )A†k , i ≤ j > k
Fi(FjAk)(F†j F†i )A†k , i > j > k
dijd†ij |0〉
= 〈0| − fjkf †jk
− FiF †i FjF †jAkA†k , i ≤ j ≤ k
FiF†i FjF
†jAkA
†k , i > j ≤ k
− FiF †i FjF †jAkA†k , i ≤ j > k
FiF†i FjF
†jAkA
†k , i > j > k
dijd†ij |0〉
∼ Sijk =
+1 , i ≤ j ≤ k
−1 , i > j ≤ k
+1 , i ≤ j > k
−1 , i > j > k
,
144
• Ai = Fi, Ak = Fk are fermions ⇒ dij = fij, djk = fjk are fermions:
Tijk ∼ 〈0| − fjkf †jk
Fi(FkAj)(F†i A†j)F
†k , i ≤ j ≤ k
Fi(FkAj)(A†jF†i )F †k , i > j ≤ k
Fi(AjFk)(F†i A†j)F
†k , i ≤ j > k
Fi(AjFk)(A†jF†i )F †k , i > j > k
fijf†ij |0〉
= 〈0| − fjkf †jk
− FiF †i AjA†jFkF †k , i ≤ j ≤ k
− FiF †i AjA†jFkF †k , i > j ≤ k
− FiF †i AjA†jFkF †k , i ≤ j > k
− FiF †i AjA†jFkF †k , i > j > k
fijf†ij |0〉
∼ Sijk = +1 ,
• Aj = Fj, Ak = Fk are fermions ⇒ dij = fij is a fermion:
Tijk ∼ 〈0| djkd†jk
Ai(FkFj)(A†iF†j )F †k , i ≤ j ≤ k
Ai(FkFj)(F†jA†i )F
†k , i > j ≤ k
Ai(FjFk)(A†iF†j )F †k , i ≤ j > k
Ai(FjFk)(F†jA†i )F
†k , i > j > k
fijf†ij |0〉
= 〈0| djkd†jk
AiA†iFjF
†j FkF
†k , i ≤ j ≤ k
AiA†iFjF
†j FkF
†k , i > j ≤ k
− AiA†iFjF †j FkF †k , i ≤ j > k
− AiA†iFjF †j FkF †k , i > j > k
fijf†ij |0〉
∼ Sijk =
+1 , i ≤ j ≤ k
+1 , i > j ≤ k
−1 , i ≤ j > k
−1 , i > j > k
,
145
• Ai = Fi, Aj = Fj, Ak = Fk are fermions:
Tijk ∼ 〈0| djkd†jk
Fi(FkFj)(F†i F†j )F †k , i ≤ j ≤ k
Fi(FkFj)(F†j F†i )F †k , i > j ≤ k
Fi(FjFk)(F†i F†j )F †k , i ≤ j > k
Fi(FjFk)(F†j F†i )F †k , i > j > k
dijd†ij |0〉
= 〈0| djkd†jk
FiF†i FjF
†j FkF
†k , i ≤ j ≤ k
− FiF †i FjF †j FkF †k , i > j ≤ k
− FiF †i FjF †j FkF †k , i ≤ j > k
FiF†i FjF
†j FkF
†k , i > j > k
dijd†ij |0〉
∼ Sijk =
+1 , i ≤ j ≤ k
−1 , i > j ≤ k
−1 , i ≤ j > k
+1 , i > j > k
.
Next, we consider the case Pi = Pj 6= Pk.
Case 2: Pi = Pj 6= Pk
The amplitude Tijk in Eq. (B.8) is for Pi = Pj 6= Pk reduced to
Tijk ∼ 〈0| dikAid†ik
(AkAi)(A†iA†k) , i ≤ k
(AiAk)(A†iA†k) , i > k
A†kdijd
†ij |0〉 , (B.9)
and the four different combinations of bosons and fermions yield the following symmetry factors:
• only bosons:
Tijk ∼ 〈0| dikd†ik
Ai(AkAi)(A
†iA†i )A
†k + Ai(AkAi)(A
†iA†i )A
†k , i ≤ k
Ai(AiAk)(A†iA†i )A
†k + Ai(AiAk)(A
†iA†i )A
†k , i > k
dijd
†ij |0〉
= 2 〈0| dikd†ik
AiA
†iAiA
†iAkA
†k , i ≤ k
AiA†iAiA
†iAkA
†k , i > k
dijd
†ij |0〉
∼ Sijk = +2 ,
146
• Ai = Fi is a fermion ⇒ dik = fik is a fermion:
Tijk ∼ 〈0| − fikf †ik
Fi(AkFi)(F
†i F†i )A†k + Fi(AkFi)(F
†i F†i )A†k , i ≤ k
Fi(FiAk)(F†i F†i )A†k + Fi(FiAk)(F
†i F†i )A†k , i > k
dijd
†ij |0〉
= 2 〈0| − fikf †ik
− FiF †i FiF †i AkA†k , i ≤ k
− FiF †i FiF †i AkA†k , i > k
dijd
†ij |0〉
∼ Sijk = +2 ,
• Ak = Fk is a fermion ⇒ dik = fik is a fermion:
Tijk ∼ 〈0| fikf †ik
Ai(FkAi)(A
†iA†i )F
†k + Ai(FkAi)(A
†iA†i )F
†k , i ≤ k
Ai(AiFk)(A†iA†i )F
†k + Ai(AiFk)(A
†iA†i )F
†k , i > k
dijd
†ij |0〉
= 2 〈0| fikf †ik
AiA
†iAiA
†iFkF
†k , i ≤ k
AiA†iAiA
†iFkF
†k , i > k
dijd
†ij |0〉
∼ Sijk = +2 ,
• Ai = Fi, Ak = Fk are fermions:
Tijk ∼ 〈0| dikd†ik
Fi(FkFi)(F
†i F†i )F †k + Fi(FkFi)(F
†i F†i )F †k , i ≤ k
Fi(FiFk)(F†i F†i )F †k + Fi(FiFk)(F
†i F†i )F †k , i > k
dijd
†ij |0〉
= 2 〈0| dikd†ik
FiF†i FiF
†i FkF
†k , i ≤ k
− FiF †i FiF †i FkF †k , i > k
dijd
†ij |0〉
∼ Sijk =
+2 , i ≤ k
−2 , i > k.
Case 3: Pi = Pk 6= Pj
We continue with the analysis of the diagram in Fig. B.3 for Pi = Pk 6= Pj. The correspondingamplitude is given by (cf. Eq. (B.8))
Tijk ∼ 〈0| dijAid†ij
(AjAi)(A†iA†j) , i ≤ j
(AiAj)(A†jA†i ) , i > j
A†idijd
†ij |0〉 . (B.10)
Consequently, we have to consider four different cases regarding the particle species:
147
• only bosons:
Tijk ∼ 〈0| dijd†ij
Ai(AjAi)(A
†iA†j)A
†i , i ≤ j
Ai(AiAj)(A†jA†i )A
†i , i > j
dijd
†ij |0〉
= 〈0| dijd†ij
AiA
†iAjA
†jAiA
†i , i ≤ j
AiA†iAjA
†jAiA
†i , i > j
dijd
†ij |0〉
∼ Sijk = +1 ,
• Ai = Fi is a fermion ⇒ dij = fij is a fermion:
Tijk ∼ 〈0| − fijf †ij
Fi(AjFi)(F
†i A†j)F
†i , i ≤ j
Fi(FiAj)(A†jF†i )F †i , i > j
fijf
†ij |0〉
= 〈0| − fijf †ij
− FiF †i AjA†jFiF †i , i ≤ j
− FiF †i AjA†jFiF †i , i > j
fijf
†ij |0〉
∼ Sijk = +1 ,
• Aj = Fj is a fermion ⇒ dij = fij is a fermion:
Tijk ∼ 〈0| fijf †ij
Ai(FjAi)(A
†iF†j )A†i , i ≤ j
Ai(AiFj)(F†jA†i )A
†i , i > j
fijf
†ij |0〉
= 〈0| fijf †ij
AiA
†iFjF
†jAiA
†i , i ≤ j
AiA†iFjF
†jAiA
†i , i > j
fijf
†ij |0〉
∼ Sijk = +1 ,
• Ai = Fi, Aj = Fj are fermions:
Tijk ∼ 〈0| dijd†ij
Fi(FjFi)(F
†i F†j )F †i , i ≤ j
Fi(FiFj)(F†j F†i )F †i , i > j
dijd
†ij |0〉
= 〈0| dijd†ij
− FiF †i FjF †j FiF †i , i ≤ j
− FiF †i FjF †j FiF †i , i > j
dijd
†ij |0〉
∼ Sijk = −1 .
148
Case 4: Pi 6= Pj = Pk
In the second to last possible configuration of identical particles where Pi 6= Pj = Pk the ampli-tude of the diagram in Fig. B.3 yields
Tijk ∼ 〈0| djkAid†jk
(AjAj)(A†iA†j) , i ≤ j
(AjAj)(A†jA†i ) , i > j
A†jdijd
†ij |0〉 , (B.11)
and the symmetry factor Sijk for different fermion / boson combinations is listed below:
• only bosons:
Tijk ∼ 〈0| djkd†jk
Ai(AjAj)(A
†iA†j)A
†j + Ai(AjAj)(A
†iA†j)A
†j , i ≤ j
Ai(AjAj)(A†jA†i )A
†j + Ai(AjAj)(A
†jA†i )A
†j , i > j
dijd
†ij |0〉
= 2 〈0| djkd†jk
AiA
†iAjA
†jAjA
†j , i ≤ j
AiA†iAjA
†jAjA
†j , i > j
dijd
†ij |0〉
∼ Sijk = +2 ,
• Ai = Fi is a fermion ⇒ dij = fij is a fermion:
Tijk ∼ 〈0| djkd†jk
Fi(AjAj)(F
†i A†j)A
†j + Fi(AjAj)(F
†i A†j)A
†j , i ≤ j
Fi(AjAj)(A†jF†i )A†j + Fi(AjAj)(A
†jF†i )A†j , i > j
fijf
†ij |0〉
= 2 〈0| djkd†jk
FiF
†i AjA
†jAjA
†j , i ≤ j
FiF†i AjA
†jAjA
†j , i > j
fijf
†ij |0〉
∼ Sijk = +2 ,
• Aj = Fj is a fermion ⇒ dij = fij is a fermion:
Tijk ∼ 〈0| djkd†jk
Ai(FjFj)(A
†iF†j )F †j + Ai(FjFj)(A
†iF†j )F †j , i ≤ j
Ai(FjFj)(F†jA†i )F
†j + Ai(FjFj)(F
†jA†i )F
†j , i > j
fijf
†ij |0〉
= 2 〈0| djkd†jk
− AiA†iFjF †j FjF †j , i ≤ j
− AiA†iFjF †j FjF †j , i > j
fijf
†ij |0〉
∼ Sijk = −2 ,
149
• Ai = Fi, Aj = Fj are fermions:
Tijk ∼ 〈0| djkd†jk
Fi(FjFj)(F
†i F†j )F †j + Fi(FjFj)(F
†i F†j )F †j , i ≤ j
Fi(FjFj)(F†j F†i )F †j + Fi(FjFj)(F
†j F†i )F †j , i > j
dijd
†ij |0〉
= 2 〈0| djkd†jk
− FiF †i FjF †j FjF †j , i ≤ j
FiF†i FjF
†j FjF
†j , i > j
dijd
†ij |0〉
∼ Sijk =
−2 , i ≤ j
+2 , i > j.
Case 5: Pi = Pj = Pk
Finally, we consider the case where all particles are identical, i.e. Pi = Pj = Pk. Hence, thereonly are two cases: all particles are either bosons or fermions. Furthermore, the correspondingamplitude,
Tijk ∼ 〈0| dAid†(AiAi)(A†iA†i )A†idd† |0〉 , (B.12)
has no explicit index dependence since for only one index there is no hierarchy left in the system(note, that we have written dij ≡ djk := d for simplicity). We find for
• only bosons:
Tijk ∼ 〈0| dd†[Ai(AiAi)(A
†iA†i )A
†i + Ai(AiAi)(A
†iA†i )A
†i
+ Ai(AiAi)(A†iA†i )A
†i + Ai(AiAi)(A
†iA†i )A
†i
]dd† |0〉
= 4 〈0| dd†AiA†iAiA†iAiA†idd† |0〉∼ Sijk = +4 ,
• only fermions (Ai = Fi):
Tijk ∼ 〈0| dd†[Fi(FiFi)(F
†i F†i )F †i + Fi(FiFi)(F
†i F†i )F †i
+ Fi(FiFi)(F†i F†i )F †i + Fi(FiFi)(F
†i F†i )F †i
]dd† |0〉
= 4 〈0| dd†[− FiF †i FiF †i FiF †i
]dd† |0〉
∼ Sijk = −4 ,
150
Summary
Collecting the results for all different configurations one can summarize them by the followingformula for the symmetry factor Sijk:
Sijk = ζijk(1 + δPiPj + δPjPk + δPiPj δPjPk
), (B.13)
where ζijk is either +1 or −1 depending on the particle species, the hierarchy between the indicesand the fact whether some of the particles are identical or not. It is defined as:
ζijk :=
−1 , if
• Pi 6= Pj 6= Pk ∧ only Pi, Pj fermions ∧i > j < k ∨ i > j > k
• Pi 6= Pj 6= Pk ∧ only Pj, Pk fermions ∧i < j > k ∨ i > j > k
• Pi 6= Pj 6= Pk ∧ Pi, Pj, Pk fermions ∧i > j < k ∨ i < j > k
• Pi = Pj 6= Pk ∧ Pi, Pj, Pk fermions ∧ i > k
• Pi = Pk 6= Pj ∧ Pi, Pj, Pk fermions .
• Pi 6= Pj = Pk ∧ only Pj fermion
• Pi 6= Pj = Pk ∧ Pi, Pj, Pk fermions ∧ i < j
• Pi = Pj = Pk ∧ Pi, Pj, Pk fermions
+1 , else
(B.14)
Note, that in each case the particles which are not mentioned to be fermions must be bosons, sothe conditions are unique.
151
Appendix C
Feynman diagrams contributing tod12–P3 dimer–particle scattering
In order to find a completely general expression for the d12–P3 scattering amplitude it is necessaryto identify all contributing Feynman diagrams. This is done in Figure C.2 and with the remarksbelow we want to motivate this result.
It is obvious that some diagrams only contribute if at least two of the three particles are identical.This is the reason for the Kronecker-delta δPiPj ,
δPiPj =
1 , if Pi = Pj, i.e. Pi and Pj are identical particles
0 , if Pi 6= Pj, i.e. Pi and Pj are distinguishable particles, (C.1)
in front of some diagrams. Moreover, we have to take into account the structure of the dimerwave function of G-parity eigenstates made of a superposition of distinguishable particles, e.g.d12 = 1√
2(A1A2 +A1A2). Choosing for example the three particle system P1 = A1, P2 = A2, P3 =
A3 = A1 the normal δP1P3 = 0 vanishes. However, since the two terms in the wave functioncan fluctuate into each other we know that the diagram in Fig. C.1 does contribute to the threeparticle system: within the blob d12 is in the state A1A2 and thus contributes together withP3 = A3 = A1 to the chosen three particle system P1 = A1, P2 = A2, P3 = A3 = A1. After theblob and during the propagation to the next vertex it fluctuates into A1A2 and decays into A1
and A2. The latter recombines with P3 = A1 to d12 which can change its state again so thatin the final state one founds the same particle content as in the initial one. Thus, a prefactorδP1P3 , which forbids such a contribution, cannot be right. For this we have introduced a modified
Kronecker-delta δ(ab(′))PiPj
which solves this inconsistency:
δ(ab(′))PiPj
:= δPiPj +(δAiAj − δPiPj
)δη(′)ab |1|
(δη(′)ab |1|− δAaAb
), (C.2)
where the first factor of the additional term ensures that nothing must changed if Pi = Pj. The
second factor checks if the dimer d(′)ab is a G-parity eigenstate and if this is true the last factor
finally checks if there is a superposition in the wave function (Aa 6= Ab) or just a single term
152
P3=A1=δP1P3
d23
A2
A1
T12
d12
A3
d12
P3d12
A2
A1
T12
d12
A1
d12
A1
iT12 ∼ δP1P3︸ ︷︷ ︸
P1=A1 6=A1=A3=P3= 0
(a)
P3=A1=δ(12)P1P3
d23
A2
A1
T12
d12
A3
d12
P3d12
A2
A1
T12
d12
A1
d12
A1
iT12 ∼ δ(12)P1P3︸ ︷︷ ︸
P1=A1 6=A1=A3=P3= 0 + (0− 1) × 1× (1− 0) = 1
(b)
Figure C.1: For d12 being a G-parity eigenstate with a superposition of states in the wave func-tion, the shown diagram contributes to the amplitude T12. However, with a ”normal“ Kronecker-delta, δP1P3 = 0, it does not (a). Since this is wrong one must introduce a modified Kronecker-
delta, δ(12)P1P3
= 1, (cf. Eq. (C.2)) to get the right result (b).
(Aa = Ab). Note, however, that not every Kronecker-delta has to be modified.Unfortunately, it is then not anymore ensured that the number of particles and of anti-particlesis conserved separately. Considering Figure C.1(b) we see that δ
(12)P1P3
= 1 also for P3 = A1 whichleads to a contradiction of particle conservation. Thus, we need in advance a prefactor a2 torestore it. In the same way we have added by hand a or b factors to some diagrams in Figure C.2.
As a second issue one observes that some diagrams become identical if two or all three particlesare identical. Thus, one would count them twice (two identical particles) or even six times (threeidentical particles). Therefore we could define a new parameter n representing the number ofidentical particles,
n =
1 , if P1 6= P2 6= P3
2 , if P1 = P2 6= P3 ∨ P1 6= P2 = P3 ∨ P1 = P3 6= P2
3 , if P1 = P2 = P3
, (C.3)
to avoid double-counting of diagrams. This straightforward procedure simply takes all diagramsinto account and divides at the end by the number of identical ones. Hence, in some sense onedoes a lot of effort to determine diagrams which are equal to others. Therefore we introducea different and a bit less straightforward approach to account for possibly identical diagrams.We do the following: whenever two amplitudes become identical because of identical particles
153
(e.g. for P1 = P2 the amplitudes T(′)13 and T
(′)23 are equal) one simply erases one of them from the
equations. Thus, there is no double-counting except for the situation that in T(′)ij the particles
Pi = Pj are identical. To correct the appearing factor of 2 we must put in by hand the additionalfactors of
(1− δPiPj/2
)in front of the diagrams. The advantage of this more complicated notation
is that one needs to calculate less parameters and thus has more clear equations.
154
Figure C.2: All diagrams contributing to a general d12–P3 scattering amplitude. The extra factors of (1−δPiPj/2) correspond
to the special counting scheme for diagrams becoming equal under some circumstances. The modified Kronecker-deltas δ(ab(′))PiPj
are defined in Eq. (C.2).
iT12
iT13
iT23
iT ′12
iT ′13
iT ′23
=
]
(1− δP1P2
2
)
[
δP1P3
δP1P2
1000
d23
d12
A3
A2
A1
δP1P3
δP1P2
1000
d23A3
A1
A2
d12
+
δP2P3
1δP1P2
000
d13
d12
A3
A1
A2
+
δP2P3a3
a2
δP1P2a2
000
d13
d12
A3
A1
A2
+
δP2P3
1δP1P2
000
d13
d12
A3
A1
A2
+
δP1P3
δP1P2
1000
d23A3
A1
A2
d12
+
δP2P3
1δP1P2
000
d13
d12
A3
A1
A2
+
δP1P3a3
δP1P2a1
a1
000
d23
d12
A3
A2
A1
+
δP2P3
1δP1P2
000
δP1P3b3δP1P2b1
b1000
d23A3
A1
A2
d12
d13
d12
A3
A1
A2
δP1P3
δP1P2
1000
d23
d12
A3
A2
A1
δP2P3
1δP1P2
000
d13
d12
A3
A1
A2
+ + ++
δP2P3
1δP1P2
000
δP1P3
δP1P2
1000
d23A3
A1
A2
d12
d13
d12
A3
A1
A2
δP1P3
δP1P2
1000
d23
d12
A3
A2
A1
δP2P3b3b2
δP1P2b2000
d13
d12
A3
A1
A2
+ + ++
000
δP2P3a3
a2
δP1P2a2
000
δP1P3
δP1P2
1
d′23A3
A1
A2
d12
d′13
d12
A3
A1
A2
000
δP1P3
δP1P2
1
d′23
d12
A3
A2
A1
000
δP2P3
1δP1P2
d′13
d12
A3
A1
A2
+ + ++
000
δP2P3
1δP1P2
000
δP1P3
δP1P2
1
d′23A3
A1
A2
d12
d′13
d12
A3
A1
A2
000
δP1P3a3
δP1P2a1
a1
d′23
d12
A3
A2
A1
000
δP2P3
1δP1P2
d′13
d12
A3
A1
A2
+ + ++
000
δP2P3
1δP1P2
000
δP1P3b3δP1P2b1
b1
d′23A3
A1
A2
d12
d′13
d12
A3
A1
A2
000
δP1P3
δP1P2
1
d′23
d12
A3
A2
A1
000
δP2P3
1δP1P2
d′13
d12
A3
A1
A2
+ + ++
000
δP2P3
1δP1P2
000
δP1P3
δP1P2
1
d′23A3
A1
A2
d12
d′13
d12
A3
A1
A2
000
δP1P3
δP1P2
1
d′23
d12
A3
A2
A1
000
δP2P3b3b2
δP1P2b2
d′13
d12
A3
A1
A2
+ + ++
155
+
000
δ(12)P1P3
δP1P2
1
d′23
A2
A1
T12
d12
A3
d12
P3
+
000
δ(12)P2P3
b3b2
δP1P2b2
d′13
A1
A2
T12
d12
A3
d12
P3
+
000
δ(12)P1P3
δP1P2
1
d′23
A2
A1
T12
d12
A3
d12
P3
+
000
δ(12)P2P3
1δP1P2
d′13
A1
A2
T12
d12
A3
d12
P3
+
000
δ(12)P1P3
δP1P2
1
d′23
A2
A1
T12
d12
A3
d12
P3
+
000
δ(12)P2P3
1δP1P2
d′13
A1
A2
T12
d12
A3
d12
P3
+
000
δ(12)P1P3
b3δP1P2b1
b1
d′23
A2
A1
T12
d12
A3
d12
P3
+
000
δ(12)P2P3
1δP1P2
d′13
A1
A2
T12
d12
A3
d12
P3
+
000
δ(12)P2P3
1δP1P2
+
000
δ(12)P1P3
δP1P2
1
+
000
δ(12)P2P3
1δP1P2
+
000
δ(12)P1P3
a3
δP1P2a1
a1
+
000
δ(12)P2P3
1δP1P2
+
000
δ(12)P1P3
δP1P2
1
+
000
δ(12)P1P3
δP1P2
1
+
000
δ(12)P2P3
a3
a2
δP1P2a2
d′23
A2
A1
T12
d12
A3
d12
P3d′13
A1
A2
T12
d12
A3
d12
P3d′23
A2
A1
T12
d12
A3
d12
P3d′13
A1
A2
T12
d12
A3
d12
P3
d′23
A2
A1
T12
d12
A3
d12
P3d′13
A1
A2
T12
d12
A3
d12
P3d′23
A2
A1
T12
d12
A3
d12
P3d′13
A1
A2
T12
d12
A3
d12
P3
+
δ(12)P1P3
δP1P2
1000
d23
A2
A1
T12
d12
A3
d12
P3
+
δ(12)P2P3
b3b2
δP1P2b2000
d13
A1
A2
T12
d12
A3
d12
P3
+
δ(12)P1P3
δP1P2
1000
d23
A2
A1
T12
d12
A3
d12
P3
+
δ(12)P2P3
1δP1P2
000
d13
A1
A2
T12
d12
A3
d12
P3
+
δ(12)P2P3
1δP1P2
000
d13
A1
A2
T12
d12
A3
d12
P3
+
δ(12)P1P3
b3δP1P2b1
b1000
d23
A2
A1
T12
d12
A3
d12
P3
+
δ(12)P2P3
1δP1P2
000
d13
A1
A2
T12
d12
A3
d12
P3
+
δ(12)P1P3
δP1P2
1000
d23
A2
A1
T12
d12
A3
d12
P3
+
δ(12)P2P3
1δP1P2
000
d13
A1
A2
T12
d12
A3
d12
P3
δ(12)P1P3
δP1P2
1000
d23
A2
A1
T12
d12
A3
d12
P3
+(1− δP1P2
2
)
[
]
+
δ(12)P1P3
a3
δP1P2a1
a1
000
d23
A2
A1
T12
d12
A3
d12
P3
+
δ(12)P2P3
1δP1P2
000
d13
A1
A2
T12
d12
A3
d12
P3
+
δ(12)P1P3
δP1P2
1000
d23
A2
A1
T12
d12
A3
d12
P3
+
δ(12)P2P3
1δP1P2
000
d13
A1
A2
T12
d12
A3
d12
P3
+
δ(12)P2P3
a3
a2
δP1P2a2
000
d13
A1
A2
T12
d12
A3
d12
P3
+
δ(12)P1P3
δP1P2
1000
d23
A2
A1
T12
d12
A3
d12
P3
156
+(1− δP1P3
2
)
[
]+
000
δP1P3
δ(13)P1P2
1
d′23
A3
A1
T13
d13
A2
d12
P3
+
000b3
δ(13)P2P3
b3δP1P3b3
d′12
A1
A3
T13
d13
A2
d12
P3
+
000
δP1P3
δ(13)P1P2
1
d′23
A3
A1
T13
d13
A2
d12
P3
+
0001
δ(13)P2P3
δP1P3
d′12
A1
A3
T13
d13
A2
d12
P3
+
000
δP1P3
δ(13)P1P2
1
d′23
A3
A1
T13
d13
A2
d12
P3
+
0001
δ(13)P2P3
δP1P3
d′12
A1
A3
T13
d13
A2
d12
P3
+
000
δP1P3b3
δ(13)P1P2
b1b1
d′23
A3
A1
T13
d13
A2
d12
P3
+
0001
δ(13)P2P3
δP1P3
d′12
A1
A3
T13
d13
A2
d12
P3
+
0001
δ(13)P2P3
δP1P3
+
000
δP1P3
δ(13)P1P2
1
+
0001
δ(13)P2P3
δP1P3
+
000
δP1P3a3
δ(13)P1P2
a1
a1
+
0001
δ(13)P2P3
δP1P3
+
000
δP1P3
δ(13)P1P2
1
+
000
δP1P3
δ(13)P1P2
1
+
000a3
δ(13)P2P3
a3
δP1P3a3
d′23
A3
A1
T13
d13
A2
d12
P3d′12
A1
A3
T13
d13
A2
d12
P3d′23
A3
A1
T13
d13
A2
d12
P3d′12
A1
A3
T13
d13
A2
d12
P3
d′23
A3
A1
T13
d13
A2
d12
P3d′12
A1
A3
T13
d13
A2
d12
P3d′23
A3
A1
T13
d13
A2
d12
P3d′12
A1
A3
T13
d13
A2
d12
P3
+
δP1P3
δ(13)P1P2
1000
d23
A3
A1
T13
d13
A2
d12
P3
+
b3
δ(13)P2P3
b3δP1P3b3
000
d12
A1
A3
T13
d13
A2
d12
P3
+
δP1P3
δ(13)P1P2
1000
d23
A3
A1
T13
d13
A2
d12
P3
+
1
δ(13)P2P3
δP1P3
000
d12
A1
A3
T13
d13
A2
d12
P3
+
1
δ(13)P2P3
δP1P3
000
d12
A1
A3
T13
d13
A2
d12
P3
+
δP1P3b3
δ(13)P1P2
b1b1000
d23
A3
A1
T13
d13
A2
d12
P3
+
1
δ(13)P2P3
δP1P3
000
d12
A1
A3
T13
d13
A2
d12
P3
+
δP1P3
δ(13)P1P2
1000
d23
A3
A1
T13
d13
A2
d12
P3
+
1
δ(13)P2P3
δP1P3
000
+
δP1P3a3
δ(13)P1P2
a1
a1
000
d23
A3
A1
T13
d13
A2
d12
P3d12
A1
A3
T13
d13
A2
d12
P3
+
δP1P3
δ(13)P1P2
1000
d23
A3
A1
T13
d13
A2
d12
P3
+
1
δ(13)P2P3
δP1P3
000
d12
A1
A3
T13
d13
A2
d12
P3
+
a3
δ(13)P2P3
a3
δP1P3a3
000
d12
A1
A3
T13
d13
A2
d12
P3
+
δP1P3
δ(13)P1P2
1000
d23
A3
A1
T13
d13
A2
d12
P3
+
1
δ(13)P2P3
δP1P3
000
d12
A1
A3
T13
d13
A2
d12
P3
δP1P3
δ(13)P1P2
1000
d23
A3
A1
T13
d13
A2
d12
P3
157
+(1− δP2P3
2
)
[
]+
000
δP2P3
1
δ(23)P1P2
d′13
A3
A2
T23
d23
A1
d12
P3
+
000b3
δP2P3b3
δ(23)P1P3
b3
d′12
A2
A3
T23
d23
A1
d12
P3
+
000
δP2P3
1
δ(23)P1P2
d′13
A3
A2
T23
d23
A1
d12
P3
+
0001
δP2P3
δ(23)P1P3
d′12
A2
A3
T23
d23
A1
d12
P3
+
000
δP2P3
1
δ(23)P1P2
d′13
A3
A2
T23
d23
A1
d12
P3
+
0001
δP2P3
δ(23)P1P3
d′12
A2
A3
T23
d23
A1
d12
P3
+
000
δP2P3b3b2
δ(23)P1P2
b2
d′13
A3
A2
T23
d23
A1
d12
P3
+
0001
δP2P3
δ(23)P1P3
d′12
A2
A3
T23
d23
A1
d12
P3
+
0001
δP2P3
δ(23)P1P3
+
000
δP2P3
1
δ(23)P1P2
+
0001
δP2P3
δ(23)P1P3
+
000
δP2P3a3
a2
δ(23)P1P2
a2
+
0001
δP2P3
δ(23)P1P3
+
000
δP2P3
1
δ(23)P1P2
+
000
δP2P3
1
δ(23)P1P2
+
000a3
δP2P3a3
δ(23)P1P3
a3
d′13
A3
A2
T23
d23
A1
d12
P3d′12
A2
A3
T23
d23
A1
d12
P3d′13
A3
A2
T23
d23
A1
d12
P3d′12
A2
A3
T23
d23
A1
d12
P3
d′13
A3
A2
T23
d23
A1
d12
P3d′12
A2
A3
T23
d23
A1
d12
P3d′13
A3
A2
T23
d23
A1
d12
P3d′12
A2
A3
T23
d23
A1
d12
P3
+
δP2P3
1
δ(23)P1P2
000
d13
A3
A2
T23
d23
A1
d12
P3
+
b3δP2P3b3
δ(23)P1P3
b3000
d12
A2
A3
T23
d23
A1
d12
P3
+
δP2P3
1
δ(23)P1P2
000
d13
A3
A2
T23
d23
A1
d12
P3
+
1δP2P3
δ(23)P1P3
000
d12
A2
A3
T23
d23
A1
d12
P3
+
1δP2P3
δ(23)P1P3
000
d12
A2
A3
T23
d23
A1
d12
P3
+
δP2P3b3b2
δ(23)P1P2
b2000
d13
A3
A2
T23
d23
A1
d12
P3
+
1δP2P3
δ(23)P1P3
000
d12
A2
A3
T23
d23
A1
d12
P3
+
δP2P3
1
δ(23)P1P2
000
d13
A3
A2
T23
d23
A1
d12
P3
+
1δP2P3
δ(23)P1P3
000
+
δP2P3a3
a2
δ(23)P1P2
a2
000
d13
A3
A2
T23
d23
A1
d12
P3d12
A2
A3
T23
d23
A1
d12
P3
+
δP2P3
1
δ(23)P1P2
000
d13
A3
A2
T23
d23
A1
d12
P3
+
1δP2P3
δ(23)P1P3
000
d12
A2
A3
T23
d23
A1
d12
P3
+
a3
δP2P3a3
δ(23)P1P3
a3
000
d12
A2
A3
T23
d23
A1
d12
P3
+
δP2P3
1
δ(23)P1P2
000
d13
A3
A2
T23
d23
A1
d12
P3
+
1δP2P3
δ(23)P1P3
000
d12
A2
A3
T23
d23
A1
d12
P3
δP2P3
1
δ(23)P1P2
000
d13
A3
A2
T23
d23
A1
d12
P3
158
+
000
δ(12′)P1P3
δP1P2
1
d′23
A2
A1
T ′12
d′12
A3
d12
P3
+
000
δ(12′)P2P3
b3b2
δP1P2b2
d′13
A1
A2
T ′12
d′12
A3
d12
P3
+
000
δ(12′)P1P3
δP1P2
1
d′23
A2
A1
T ′12
d′12
A3
d12
P3
+
000
δ(12′)P2P3
1δP1P2
d′13
A1
A2
T ′12
d′12
A3
d12
P3
+
000
δ(12′)P1P3
δP1P2
1
d′23
A2
A1
T ′12
d′12
A3
d12
P3
+
000
δ(12′)P2P3
1δP1P2
d′13
A1
A2
T ′12
d′12
A3
d12
P3
+
000
δ(12′)P1P3
b3δP1P2b1
b1
d′23
A2
A1
T ′12
d′12
A3
d12
P3
+
000
δ(12′)P2P3
1δP1P2
d′13
A1
A2
T ′12
d′12
A3
d12
P3
+
000
δ(12′)P2P3
1δP1P2
+
000
δ(12′)P1P3
δP1P2
1
+
000
δ(12′)P2P3
1δP1P2
+
000
δ(12′)P1P3
a3
δP1P2a1
a1
+
000
δ(12′)P2P3
1δP1P2
+
000
δ(12′)P1P3
δP1P2
1
+
000
δ(12′)P1P3
δP1P2
1
+
000
δ(12′)P2P3
a3
a2
δP1P2a2
d′23
A2
A1
T ′12
d′12
A3
d12
P3d′13
A1
A2
T ′12
d′12
A3
d12
P3d′23
A2
A1
T ′12
d′12
A3
d12
P3d′13
A1
A2
T ′12
d′12
A3
d12
P3
d′23
A2
A1
T ′12
d′12
A3
d12
P3d′13
A1
A2
T ′12
d′12
A3
d12
P3d′23
A2
A1
T ′12
d′12
A3
d12
P3d′13
A1
A2
T ′12
d′12
A3
d12
P3
+
δ(12′)P1P3
δP1P2
1000
d23
A2
A1
T ′12
d′12
A3
d12
P3
+
δ(12′)P2P3
b3b2
δP1P2b2000
d13
A1
A2
T ′12
d′12
A3
d12
P3
+
δ(12′)P1P3
δP1P2
1000
d23
A2
A1
T ′12
d′12
A3
d12
P3
+
δ(12′)P2P3
1δP1P2
000
d13
A1
A2
T ′12
d′12
A3
d12
P3
+
δ(12′)P2P3
1δP1P2
000
d13
A1
A2
T ′12
d′12
A3
d12
P3
+
δ(12′)P1P3
b3δP1P2b1
b1000
d23
A2
A1
T ′12
d′12
A3
d12
P3
+
δ(12′)P2P3
1δP1P2
000
d13
A1
A2
T ′12
d′12
A3
d12
P3
+
δ(12′)P1P3
δP1P2
1000
d23
A2
A1
T ′12
d′12
A3
d12
P3
+
δ(12′)P2P3
1δP1P2
000
+
δ(12′)P1P3
a3
δP1P2a1
a1
000
d23
A2
A1
T ′12
d′12
A3
d12
P3d13
A1
A2
T ′12
d′12
A3
d12
P3
+
δ(12′)P1P3
δP1P2
1000
d23
A2
A1
T ′12
d′12
A3
d12
P3
+
δ(12′)P2P3
1δP1P2
000
d13
A1
A2
T ′12
d′12
A3
d12
P3
+
δ(12′)P2P3
a3
a2
δP1P2a2
000
d13
A1
A2
T ′12
d′12
A3
d12
P3
+
δ(12′)P1P3
δP1P2
1000
d23
A2
A1
T ′12
d′12
A3
d12
P3
+
δ(12′)P2P3
1δP1P2
000
d13
A1
A2
T ′12
d′12
A3
d12
P3
δ(12′)P1P3
δP1P2
1000
d23
A2
A1
T ′12
d′12
A3
d12
P3
+(1− δP1P2
2
)
[
]
159
+(1− δP1P3
2
)
[
]+
000
δP1P3
δ(13′)P1P2
1
d′23
A3
A1
T ′13
d′13
A2
d12
P3
+
000b3
δ(13′)P2P3
b3δP1P3b3
d′12
A1
A3
T ′13
d′13
A2
d12
P3
+
000
δP1P3
δ(13′)P1P2
1
d′23
A3
A1
T ′13
d′13
A2
d12
P3
+
0001
δ(13′)P2P3
δP1P3
d′12
A1
A3
T ′13
d′13
A2
d12
P3
+
000
δP1P3
δ(13′)P1P2
1
d′23
A3
A1
T ′13
d′13
A2
d12
P3
+
0001
δ(13′)P2P3
δP1P3
d′12
A1
A3
T ′13
d′13
A2
d12
P3
+
000
δP1P3b3
δ(13′)P1P2
b1b1
d′23
A3
A1
T ′13
d′13
A2
d12
P3
+
0001
δ(13′)P2P3
δP1P3
d′12
A1
A3
T ′13
d′13
A2
d12
P3
+
0001
δ(13′)P2P3
δP1P3
+
000
δP1P3
δ(13′)P1P2
1
+
0001
δ(13′)P2P3
δP1P3
+
000
δP1P3a3
δ(13′)P1P2
a1
a1
+
0001
δ(13′)P2P3
δP1P3
+
000
δP1P3
δ(13′)P1P2
1
+
000
δP1P3
δ(13′)P1P2
1
+
000a3
δ(13′)P2P3
a3
δP1P3a3
d′23
A3
A1
T ′13
d′13
A2
d12
P3d′12
A1
A3
T13
d′13
A2
d12
P3d′23
A3
A1
T ′13
d′13
A2
d12
P3d′12
A1
A3
T ′13
d′13
A2
d12
P3
d′23
A3
A1
T ′13
d′13
A2
d12
P3d′12
A1
A3
T ′13
d′13
A2
d12
P3d′23
A3
A1
T ′13
d′13
A2
d12
P3d′12
A1
A3
T ′13
d′13
A2
d12
P3
+
δP1P3
δ(13′)P1P2
1000
d23
A3
A1
T ′13
d′13
A2
d12
P3
+
b3
δ(13′)P2P3
b3δP1P3b3
000
d12
A1
A3
T ′13
d′13
A2
d12
P3
+
δP1P3
δ(13′)P1P2
1000
d23
A3
A1
T ′13
d′13
A2
d12
P3
+
1
δ(13′)P2P3
δP1P3
000
d12
A1
A3
T ′13
d′13
A2
d12
P3
+
1
δ(13′)P2P3
δP1P3
000
d12
A1
A3
T ′13
d′13
A2
d12
P3
+
δP1P3b3
δ(13′)P1P2
b1b1000
d23
A3
A1
T ′13
d′13
A2
d12
P3
+
1
δ(13′)P2P3
δP1P3
000
d12
A1
A3
T ′13
d′13
A2
d12
P3
+
δP1P3
δ(13′)P1P2
1000
d23
A3
A1
T ′13
d′13
A2
d12
P3
+
1
δ(13′)P2P3
δP1P3
000
+
δP1P3a3
δ(13′)P1P2
a1
a1
000
d23
A3
A1
T ′13
d′13
A2
d12
P3d12
A1
A3
T ′13
d′13
A2
d12
P3
+
δP1P3
δ(13′)P1P2
1000
d23
A3
A1
T ′13
d′13
A2
d12
P3
+
1
δ(13′)P2P3
δP1P3
000
d12
A1
A3
T ′13
d′13
A2
d12
P3
+
a3
δ(13′)P2P3
a3
δP1P3a3
000
d12
A1
A3
T ′13
d′13
A2
d12
P3
+
δP1P3
δ(13′)P1P2
1000
d23
A3
A1
T ′13
d′13
A2
d12
P3
+
1
δ(13′)P2P3
δP1P3
000
d12
A1
A3
T ′13
d′13
A2
d12
P3
δP1P3
δ(13′)P1P2
1000
d23
A3
A1
T ′13
d′13
A2
d12
P3
160
+(1− δP2P3
2
)
[
]+
000
δP2P3
1
δ(23′)P1P2
d′13
A3
A2
T ′23
d′23
A1
d12
P3
+
000b3
δP2P3b3
δ(23′)P1P3
b3
d′12
A2
A3
T ′23
d′23
A1
d12
P3
+
000
δP2P3
1
δ(23′)P1P2
d′13
A3
A2
T ′23
d′23
A1
d12
P3
+
0001
δP2P3
δ(23′)P1P3
d′12
A2
A3
T ′23
d′23
A1
d12
P3
+
000
δP2P3
1
δ(23′)P1P2
d′13
A3
A2
T ′23
d′23
A1
d12
P3
+
0001
δP2P3
δ(23′)P1P3
d′12
A2
A3
T ′23
d′23
A1
d12
P3
+
000
δP2P3b3b2
δ(23′)P1P2
b2
d′13
A3
A2
T ′23
d′23
A1
d12
P3
+
0001
δP2P3
δ(23′)P1P3
d′12
A2
A3
T ′23
d′23
A1
d12
P3
+
0001
δP2P3
δ(23′)P1P3
+
000
δP2P3
1
δ(23′)P1P2
+
0001
δP2P3
δ(23′)P1P3
+
000
δP2P3a3
a2
δ(23′)P1P2
a2
+
0001
δP2P3
δ(23′)P1P3
+
000
δP2P3
1
δ(23′)P1P2
+
000
δP2P3
1
δ(23′)P1P2
+
000a3
δP2P3a3
δ(23′)P1P3
a3
d′13
A3
A2
T ′23
d23
A1
d12
P3d′12
A2
A3
T ′23
d′23
A1
d12
P3d′13
A3
A2
T ′23
d′23
A1
d12
P3d′12
A2
A3
T ′23
d′23
A1
d12
P3
d′13
A3
A2
T ′23
d′23
A1
d12
P3d′12
A2
A3
T ′23
d′23
A1
d12
P3d′13
A3
A2
T ′23
d′23
A1
d12
P3d′12
A2
A3
T ′23
d′23
A1
d12
P3
+
δP2P3
1
δ(23′)P1P2
000
d13
A3
A2
T ′23
d′23
A1
d12
P3
+
b3δP2P3b3
δ(23′)P1P3
b3000
d12
A2
A3
T ′23
d′23
A1
d12
P3
+
δP2P3
1
δ(23′)P1P2
000
d13
A3
A2
T ′23
d′23
A1
d12
P3
+
1δP2P3
δ(23′)P1P3
000
d12
A2
A3
T ′23
d′23
A1
d12
P3
+
1δP2P3
δ(23′)P1P3
000
d12
A2
A3
T ′23
d′23
A1
d12
P3
+
δP2P3b3b1
δ(23′)P1P2
b1000
d13
A3
A2
T ′23
d′23
A1
d12
P3
+
1δP2P3
δ(23′)P1P3
000
d12
A2
A3
T ′23
d′23
A1
d12
P3
+
δP2P3
1
δ(23′)P1P2
000
d13
A3
A2
T ′23
d′23
A1
d12
P3
+
1δP2P3
δ(23′)P1P3
000
+
δP2P3a3
a2
δ(23′)P1P2
a2
000
d13
A3
A2
T ′23
d′23
A1
d12
P3d12
A2
A3
T ′23
d′23
A1
d12
P3
+
δP2P3
1
δ(23′)P1P2
000
d13
A3
A2
T ′23
d′23
A1
d12
P3
+
1δP2P3
δ(23′)P1P3
000
d12
A2
A3
T ′23
d′23
A1
d12
P3
+
a3
δP2P3a3
δ(23′)P1P3
a3
000
d12
A2
A3
T ′23
d′23
A1
d12
P3
+
δP2P3
1
δ(23′)P1P2
000
d13
A3
A2
T ′23
d′23
A1
d12
P3
+
1δP2P3
δ(23′)P1P3
000
d12
A2
A3
T ′23
d′23
A1
d12
P3
δP2P3
1
δ(23′)P1P2
000
d13
A3
A2
T ′23
d′23
A1
d12
P3
161
Appendix D
Projection on L-th partial wave
A scattering amplitude T (E,k,p) can be expanded in partial waves as
T (E,k,p) =∞∑
L=0
(2L+ 1) T (L)(E, k, p)PL(cos θ) , (D.1)
with the Legendre polynomial (i.e. Legendre function of the first kind) PL and θ = ^(k,p) beingthe angle between incoming and outgoing momenta whose moduli we write as k and p. Sincethe Legendre polynomials are orthogonal,
∫ 1
−1
d cos θ PL(cos θ)P`(cos θ) =2
2L+ 1δL` , (D.2)
one can project out the L-th partial wave using an operator defined via
T (L)(E, k, p) =1
2
∫ 1
−1
d cos(θ) PL(cos θ)T (E,k,p) . (D.3)
In Fig. D.1 we have defined the angles between the momenta of a scattering amplitude T (E,k,p).Following Ref. [163] in this system it holds the useful relation
PL(cosψ) =4π
2L+ 1
L∑
m=−LY ∗Lm (θ′, ϕ′)YLm (θ, ϕ)
= PL(cos θ)PL (cos θ′) + 2L∑
m=1
(L−m)!
(L+m)!PmL (cos θ)PmL (cos θ′) cos [m (ϕ− ϕ′)] , (D.4)
where YLm are the spherical harmonics and PmL are the associated Legendre polynomials.
In the following we will project out the L-th partial wave of the d(′)ij –Pk scattering amplitudes
contributing to Eq. (F.2). For the moment we will ignore all terms which do not depend on the
162
~p~q
~k
z
y
x
θ′
θ
ψ
ϕ
ϕ′
Figure D.1: Definition of angles between the different momenta of a scattering amplitudeT (E,k,p) where we have chosen p so that it points in z direction.
momenta k, p, q or their moduli k, p, q and write
T(′)ij (E,k,p) ∼ 1
E − k2
2m3− p2
2m1− (k+p)2
2m2+ iε
+1
E − k2
2m3− p2
2m2− (k+p)2
2m1+ iε
+3∑
i, j, k = 1i < jk 6= i, j
∫d3q
(2π)3
Tij(E,k,q)
−γij +
√−2µij
(E − q2
2mk− q2
2(mi+mj)
)− iε
×
1
E − q2
2mk− p2
2mi− (q+p)2
2mj+ iε
+1
E − q2
2mk− p2
2mj− (q+p)2
2mi+ iε
+3∑
i, j, k = 1i < jk 6= i, j
∫d3q
(2π)3
T ′ij(E,k,q)
−γ′ij +
√−2µij
(E − q2
2mk− q2
2(mi+mj)
)− iε
×
1
E − q2
2mk− p2
2mi− (q+p)2
2mj+ iε
+1
E − q2
2mk− p2
2mj− (q+p)2
2mi+ iε
.
(D.5)
If one applies the projection operator Eq. (D.3) and define according to Fig. D.1 the anglesθ = ^ (k,p), θ′ = ^ (q,p) and ψ = ^ (k,q) one can rewrite the d3q integral in spherical
163
coordinates q, θ′, ϕ′ so that Eq. (D.5) reads
T(′)(L)ij (E, k, p) =
1
2
∫ 1
−1
d cos θ PL(cos θ)T(′)ij (E,k,p) ∼
1
2
∫ 1
−1
d cos θPL(cos θ)
E − k2
2µ23− p2
2µ12− kp
m2cos θ + iε
+1
2
∫ 1
−1
d cos θPL(cos θ)
E − k2
2µ13− p2
2µ12− kp
m1cos θ + iε
+3∑
i, j, k = 1i < jk 6= i, j
1
(2π)3
1
2
∫ 1
−1
d cos θPL(cos θ)
∫ ∞
0
dq
∫ 1
−1
d cos θ′∫ 2π
0
dϕ′
×
q2∑∞
`=0(2`+ 1)T(`)ij (E, k, q)P`(cosψ)
−γij +
√−2µij
(E − q2
2mk− q2
2(mi+mj)
)− iε
+q2∑∞
`=0(2`+ 1)T′(`)ij (E, k, q)P`(cosψ)
−γ′ij +
√−2µij
(E − q2
2mk− q2
2(mi+mj)
)− iε
×(
1
E − q2
2µkj− p2
2µij− qp
mjcos θ′ + iε
+1
E − q2
2µki− p2
2µij− qp
micos θ′ + iε
), (D.6)
where Eq. (D.1) was used to replace the amplitudes on the right-hand-side by their partial waveexpansions and µij is the reduced mass defined in Eq. (3.34).Consider the dϕ′ integral. From Eq. (D.4) we know that P`(cosψ) depends on ϕ′. In fact, it isthe only ϕ′ dependent term in the integral and hence we find
∫ 2π
0
dϕ′P`(cosψ) = P`(cos θ)P`(cos θ′)
∫ 2π
0
dϕ′
+ 2∑
m=1
(`−m)!
(`+m)!Pm` (cos θ)Pm` (cos θ′)
∫ 2π
0
dϕ′ cos [m (ϕ− ϕ′)]︸ ︷︷ ︸
= 0 ∀m ∈ N= 2πP`(cos θ)P`(cos θ′) . (D.7)
Thus, the second term of Eq. (D.6) is now proportional to the product PL(cos θ)P`(cos θ). To-gether with the d cos θ integration the mentioned product yields due to the orthogonality relation
164
Eq. (D.2) just a factor of 2δL`/(2L+ 1) and one ends up with
T(′)(L)ij (E, k, p) ∼ − m2
kp
1
2
∫ 1
−1
d cos θPL(cos θ)
m2
kp
(k2
2µ23+ p2
2µ12− E
)+ cos θ − iε
− m1
kp
1
2
∫ 1
−1
d cos θPL(cos θ)
m1
kp
(k2
2µ13+ p2
2µ12− E
)+ cos θ − iε
− 1
2π2
3∑
i, j, k = 1i < jk 6= i, j
∫ ∞
0
dq
q2T(L)ij (E, k, q)
−γij +
√−2µij
(E − q2
2mk− q2
2(mi+mj)
)− iε
+q2T
′(L)ij (E, k, q)
−γ′ij +
√−2µij
(E − q2
2mk− q2
2(mi+mj)
)− iε
×
mj
qp
1
2
∫ 1
−1
d cos θ′PL(cos θ′)
mjqp
(q2
2µkj+ p2
2µij− E
)+ cos θ′ − iε
+mi
qp
1
2
∫ 1
−1
d cos θ′PL(cos θ′)
miqp
(q2
2µki+ p2
2µij− E
)+ cos θ′ − iε
. (D.8)
Now the question arises how one deals with the remaining d cos θ(′) integrals proportional to theLegendre polynomials? For this purpose we firstly define a short-hand-notation for this kind ofintegral:
QL(β − iε) := (−1)L1
2
∫ 1
−1
dxPL(x)
(β − iε) + x, with β ∈ R \ −1,+1 . (D.9)
The restriction that |β| 6= 1 is necessary since at these two points QL(β − iε) becomes singular.Note, that Eq. (D.9) coincides with the definition of the Legendre function of the second kindwith complex argument introduced in Ref. [164]; we will point out this connection below in moredetail.Motivated by the fact that we assume low energy scattering of P3 off the dimer d12, one can argue– similarly to section 3.1.3 where the S-wave dimer approach was justified – that also d12–P3
scattering is dominated by S-wave interactions. Therefore it is a convenient method [80] to setL = 0 which simplifies Eq. (D.9) according to P0(x) = 1 ∀ x ∈ R to
Q0(β − iε) =1
2
∫ 1
−1
dx1
(β − iε) + x=
1
2
[ln (β + 1− iε)− ln (β − 1− iε)
]. (D.10)
In the limit ε→ 0 we have to distinguish three cases for β 6= ±1:
165
• −1 < β < 1: ⇒ β + 1 = |β + 1| ∧ β − 1 = −|β − 1|, with |β ± 1| ∈ R+
⇒ limε→0
Q0(β − iε) =1
2
[ln (|β + 1|)− ln (−|β − 1|)
]
=1
2
[ln (|β + 1|)− ln (|β − 1|)− i arg(−|β − 1|)
]
=1
2
[ln (|β + 1|)− ln (|β − 1|)− iπ
]
=1
2
[ln
( |β + 1||β − 1|
)− iπ
]=
1
2
[ln
(∣∣∣∣β + 1
β − 1
∣∣∣∣)− iπ
], (D.11)
• β < 1: ⇒ β ± 1 = −|β ± 1|, with |β ± 1| ∈ R+
⇒ limε→0
Q0(β − iε) =1
2
[ln (−|β + 1|)− ln (−|β − 1|)
]
=1
2
[ln (|β + 1|) + i arg(−|β + 1|)− ln (|β − 1|)− i arg(−|β − 1|)
]
=1
2
[ln (|β + 1|) + iπ − ln (|β − 1|)− iπ
]
=1
2
[ln
( |β + 1||β − 1|
)]=
1
2
[ln
(∣∣∣∣β + 1
β − 1
∣∣∣∣)]
, (D.12)
• β > 1: ⇒ β ± 1 = |β ± 1|, with |β ± 1| ∈ R+
⇒ limε→0
Q0(β − iε) =1
2
[ln (−|β + 1|)− ln (−|β − 1|)
]
=1
2
[ln (|β + 1|)− ln (|β − 1|)
]
=1
2
[ln
( |β + 1||β − 1|
)]=
1
2
[ln
(∣∣∣∣β + 1
β − 1
∣∣∣∣)]
. (D.13)
Here, we have used that the logarithm of a complex number can be written as
ln(−x) = ln(| − x|) + i arg(−x) = ln(x) + iπ , x ∈ R+ . (D.14)
In summary we thus find
limε→0
Q0(β − iε) =
12
[ln(∣∣∣β+1
β−1
∣∣∣)− iπ
], if |β| < 1
12
ln(∣∣∣β+1
β−1
∣∣∣), else
. (D.15)
This result could then be plugged into Eq. (D.8) so that one could determine the d12–P3 S-wavescattering amplitude which corresponds to an approximation sufficient for most three particle
166
systems. However, ignoring this fact for the moment, one can in the same way determine thelimit of Q1(β − iε) for ε going to zero:
limε→0
Q1(β − iε) = limε→0
1
2
∫ 1
−1
dxx
(β − iε) + x
= limε→0
1
2(β − iε)
[ln (β + 1− iε)− ln (β − 1− iε)
]− 1
=
β2
[ln(∣∣∣β+1
β−1
∣∣∣)− iπ
]− 1 , if |β| < 1
β2
ln(∣∣∣β+1
β−1
∣∣∣)− 1 , else
. (D.16)
Already at this point we notice that – especially due to the (−1)L factor in Eq. (D.9) – bothresults for L = 0 and for L = 1 are quite similar to the first two Legendre functions of the secondkind, which we call QL. In fact, the only difference between QL and QL is that the denominatorin the logarithm is not |β − 1|, but |1 − β|. From this we conclude that the standard Legendrefunctions of the second kind are real valued for |β| < 1, but complex outside the interval [−1, 1];in contrast to QL where the situation is interchanged (note, that the singularities at ±1 are notaffected). However, this difference is not surprising since the Legendre differential equation,
d
dx
[(1− x2
) dy(x)
dx
]+ L(L+ 1)y(x) = 0 , with L ∈ N+ and |x| < 1 , (D.17)
with two linearly independent solutions y(x) = APL(x) +BQL(x) is only defined on the interval
[−1, 1]. Therefore one chooses the brunch cut of the complex logarithm in QL so that it is realinside this interval. Doing it the other way around (complex inside the interval [−1, 1]) one endsup with QL. Consequently, we conclude that QL are the Legendre functions of the second kindwith a non-standard convention regarding the brunch cut of the complex logarithm [164]. Wewill thus also refer to QL as Legendre function of the second kind. If necessary we would add theterm ”standard“ (”non-standard“) Legendre function of the second kind if QL (QL) is meant (asit was done above).Although we have already found an explicit expression for Q0 and Q1 and know from the low-energy scattering approximation that higher partial waves need not to be taken into account,we will keep the general QL in our scattering amplitude. This is done on the one hand becausewe will later on derive results where the explicit form of QL leads to integrals which cannotbe straightforwardly calculated anyway. On the other hand because we simply want to claim ahigh level of generality in this work. Therefore we end up with a partial wave projected d
(′)ij –Pk
167
scattering amplitude given by
T(′)(L)ij (E, k, p) ∼
− m2
kpQL
(m2
kp
(k2
2µ23
+p2
2µ12
− E)− iε
)− m1
kpQL
(m1
kp
(k2
2µ13
+p2
2µ12
− E)− iε
)
− 1
2π2
3∑
i, j, k = 1i < jk 6= i, j
∫ ∞
0
dq
q2T(L)ij (E, k, q)
−γij +
√−2µij
(E − q2
2mk− q2
2(mi+mj)
)− iε
+q2T
′(L)ij (E, k, q)
−γ′ij +
√−2µij
(E − q2
2mk− q2
2(mi+mj)
)− iε
×[mj
qpQL
(mj
qp
(q2
2µkj+
p2
2µij− E
)− iε
)+mi
qpQL
(mi
qp
(q2
2µki+
p2
2µij− E
)− iε
)],
(D.18)
and thus Eq. (F.2) simplifies to Eq. (F.3) shown in appendix F.
168
Appendix E
Mellin transform of Legendre functionsof the second kind
In this section we want to calculate the integral
∫ ∞
0
dXXs(L)−1QL
(αX + β
1
X
), (E.1)
which repeatedly appears in the derivation of the transcendental equations for type 1, type 2and type 3 systems. For this purpose we closely follow the work of Grießhammer in Ref. [165].Hence, we firstly notice that the Mellin transform of a function f(z) is defined as [168]
M [f(z), s] :=
∫ ∞
0
dzzs−1f(z) . (E.2)
Therefore Eq. (E.1) is nothing else then the Mellin transform of the Legendre function of thesecond kind:
M[QL
(αX + β
1
X
), s(L)
]:=
∫ ∞
0
dXX(L)−1QL
(αX + β
1
X
). (E.3)
Comparing our definition of QL (see Eq. (D.9)),
QL(z) := (−1)L1
2
∫ 1
−1
dxPL(x)
z + x,
with that of Grießhammer (cf. Eq. (2.4) in Ref. [165]),
Q(HG)L (z) :=
1
2
∫ 1
−1
dxPL(x)
z − x ,
we observe that
QL(z) = (−1)L+1Q(HG)L (−z) . (E.4)
169
However, in Eq. (A.7) in the appendix of Ref. [165] it is shown that Q(HG)L can be expressed
in terms of hypergeometric functions which additionally are expressed in a series representation[164]. Using the same method we find for QL the following result:
QL(z) =
√πΓ(L+ 1)
2L+1Γ(L2
+ 1)
Γ(L+1
2
)∞∑
n=0
Γ(L2
+ 1 + n)
Γ(L+1
2+ n)
Γ(L+ 3
2+ n)
Γ (n+ 1)(−1)−2nz−(2n+L+1) , (E.5)
which is indeed equivalent to the representation of Q(HG)L (z) since (−1)−2n = 1 for all n ∈ N.
However, in contrast to Ref. [165] we consider the argument z = αX + βX−1. Thus, the Mellintransform in Eq. (E.3) is given as
M[QL
(αX + β
1
X
), s(L)
]=
√π Γ(L+ 1)
2L+1Γ(L2
+ 1)
Γ(L+1
2
)∞∑
n=0
Γ(L2
+ 1 + n)
Γ(L+1
2+ n)
Γ(L+ 3
2+ n)
Γ (n+ 1)
× α−(2n+L+1)
∫ ∞
0
dXX2n+L+s(L)
(X2 +
β
α
)−(2n+L+1)
. (E.6)
As it is done in Ref. [165] the integral over X can be solved using Ref. [164] or even withMathematica:
∫ ∞
0
dXX2n+L+s(L)
(X2 +
β
α
)−(2n+L+1)
=1
2
(β
α
) 12
(−1−L−n2+s(L))
×Γ(n+ L+s(L)+1
2
)Γ(n+ L−s(L)+1
2
)
Γ (2n+ L+ 1), (E.7)
if β/α > 0, 1 +L+ 2n > Re(s(L)) and 1 +L+ 2n+ Re(s(L)) > 0 holds. Note, that since L, n ≥ 0the second condition (1 +L+ 2n > Re(s(L))) directly implies the third and additionally we knowthat for µ and µ being placeholders for various combinations of µij and µik it holds
β
α=m
2µ
2µ
m=µ
µ> 0 , ∀ µ, µ . (E.8)
Finally, it is shown in Ref. [165] that the remaining condition 1 + L + 2n > Re(s(L)) is alsofulfilled. Thus, it is indeed allowed to write the Mellin transform Eq. (E.6) as
M[QL
(αX + β
1
X
), s(L)
]=
1
2
√π Γ(L+ 1)
2L+1Γ(L2
+ 1)
Γ(L+1
2
)∞∑
n=0
Γ(L2
+ 1 + n)
Γ(L+1
2+ n)
Γ(L+ 3
2+ n)
Γ (n+ 1)
×(
1√αβ
)2n+L+1(β
α
) s(L)
2 Γ(n+ L+s(L)+1
2
)Γ(n+ L−s(L)+1
2
)
Γ (2n+ L+ 1).
(E.9)
In the next step we use the well-known identity Γ(z + 1) = z Γ(z) and the Legendre doublingformula,
Γ(z
2
)Γ
(z + 1
2
)=
π
2z−1Γ(z) , ∀ z ∈ C \ 0,−1,−2, ... , (E.10)
170
to simplify the term
I :=Γ(L+ 1)
2L+1Γ(L2
+ 1)
Γ(L+1
2
) Γ(L2
+ 1 + n)
Γ(L+1
2+ n)
Γ (2n+ L+ 1), (E.11)
appearing in the Mellin transform above. Since Eq. (E.10) is only valid for z < 0 we can use itsolely in the case of L > 0. Then it holds:
• Γ(L2
+ 1)
Γ(L+1
2
)= L
2Γ(L2
)Γ(L+1
2
)= L
√π
2LΓ(L),
• Γ(L2
+ 1 + n)
Γ(L+1
2+ n)
= L+2n2
Γ(L+2n
2
)Γ(L+2n+1
2
)= (L+ 2n)
√π
2L+2nΓ (L+ 2n),
and thus I = 2−(L+1) 4−n. Plugging this result for I into Eq. (E.9) one finds for L > 0
M[QL
(αX + β
1
X
), s(L)
]=
√π
2L+2
(β
α
) s(L)
2(
1√αβ
)L+1
×∞∑
n=0
Γ(n+ L+s(L)+1
2
)Γ(n+ L−s(L)+1
2
)
Γ(L+ 3
2+ n)
Γ (n+ 1)(4αβ)−n . (E.12)
Using the series representation of the hypergeometric functions pFq [164],
pFq(a1, ..., ap; b1, ..., bq; z) =∞∑
k=0
p∏
i=1
Γ(k + ai)
Γ(ai)
q∏
j=1
Γ(bj)
Γ(k + bj)
zk
k!, p, q ∈ N , (E.13)
one concludes that also Eq. (E.12) can be written in terms of 2F1:
M[QL
(αX + β
1
X
), s(L)
]=
√π
2L+2
(β
α
) s(L)
2(
1√αβ
)L+1 Γ(L+s(L)+1
2
)Γ(L−s(L)+1
2
)
Γ(
2L+32
)
× 2F1
(L+ s(L) + 1
2,L− s(L) + 1
2;2L+ 3
2;
1
4αβ
), (E.14)
valid for L > 0.In order to show that this result also holds for S-waves we insert L = 0 explicitly into Eq. (E.9)and use Γ(1) = 1, Γ(1/2) =
√π and Γ(z + 1) = z Γ(z) to deduce
M[Q0
(αX + β
1
X
), s(0)
]=
√π
2
1√αβ
(β
α
) s(0)
2 Γ(s(0)+1
2
)Γ(−s(0)+1
2
)
Γ(
32
)
×∞∑
n=0
Γ(n+ s(0)+1
2
)Γ(n+ −s(0)+1
2
)
Γ(s(0)+1
2
)Γ(−s(0)+1
2
) Γ(
32
)
Γ(n+ 3
2
)(
1
4αβ
)n1
n!
× 4nΓ(n+ 1)Γ(n+ 1
2
)
Γ(2n+ 1)Γ(
12
) . (E.15)
171
For n = 0 we observe that the last term in the equation above yields 1, but also for n > 0 onecan show with the Legendre doubling formula that
4nΓ(n+ 1)Γ(n+ 1
2
)
Γ(2n+ 1)Γ(
12
) n6=0=
4nnΓ(n)Γ(n+ 1
2
)
2nΓ(2n)Γ(
12
) =4nΓ
(2n2
)Γ(
2n+12
)
2Γ(2n)√π
n6=0=
4n√π 2−(2n−1)Γ(2n)
2Γ(2n)√π
=4n
22n−1 × 2= 1 . (E.16)
Therefore Eq. (E.15) is equivalent to
M[Q0
(αX + β
1
X
), s(0)
]=
√π
2
1√αβ
(β
α
) s(0)
2 Γ(s(0)+1
2
)Γ(−s(0)+1
2
)
Γ(
32
)
× 2F1
(s(0) + 1
2,−s(0) + 1
2;3
2;
1
4αβ
), (E.17)
and we conclude that the relation
M[QL
(αX + β
1
X
), s(L)
]=
√π
2L+2
(β
α
) s(L)
2(
1√αβ
)L+1 Γ(L+s(L)+1
2
)Γ(L−s(L)+1
2
)
Γ(
2L+32
)
× 2F1
(L+ s(L) + 1
2,L− s(L) + 1
2;2L+ 3
2;
1
4αβ
), (E.18)
holds for all partial waves L ≥ 0.In particular, we can use the FunctionExpand command implemented in Mathematica to writeEq. (E.18) for L = 0 and for L = 1 in terms of trigonometric functions. The following identitieshold in the S-wave case:
• Γ(L+s(L)+1
2
)Γ(L−s(L)+1
2
)∣∣∣L=0
= Γ(s(0)+1
2
)Γ(−s(0)+1
2
)= π
cos(π2 s(0)),
• Γ(
2L+32
)∣∣L=0
= Γ(
32
)=√π
2,
• 2F1
(L+s(L)+1
2, L−s
(L)+12
; 2L+32
; z)∣∣∣
L=0= 2F1
(s(0)+1
2, −s
(0)+12
; 32; z)
=sin(s(0) arcsin(
√z))
s(0)√z
.
Also for the P -wave case the FunctionExpand command provides:
• Γ(L+s(L)+1
2
)Γ(L−s(L)+1
2
)∣∣∣L=1
= Γ(
1+s(1)+12
)Γ(
1−s(1)+12
)= πs(1)
21
sin(π2 s(1)),
• Γ(
2L+32
)∣∣L=1
= Γ(1 + 3
2
)= 3
√π
4,
• 2F1
(L+s(L)+1
2, L−s
(L)+12
; 2L+32
; z)∣∣∣
L=1= 2F1
(1+s(1)+1
2, 1−s(1)+1
2; 2+3
2; z)
g = 3
[(s(1))2−1]z
[ √1−z
s(1)√z
sin
(s(1) arcsin (
√z)
)− cos
(s(1) arcsin (
√z)
)].
172
Therefore we find for L = 0 that the Mellin transform of Legendre function of the second kindis given by
M[Q0
(αX + β
1
X
), s(0)
]=
∫ ∞
0
dX Xs(0)i −1Q0
(αX + β
1
X
)
=
(√β
α
)s(0)
π
s(0)
sin(s(0) arcsin
(12
√1αβ
))
cos(π2s(0)) , (E.19)
and similarly for L = 1:
M[Q1
(αX + β
1
X
), s(1)
]=
∫ ∞
0
dX Xs(1)−1Q1
(αX + β
1
X
)
=
(√β
α
)s(1)
πs(1)
[(s(1))
2 − 1] 1
sin(π2s(1))
×[√
4αβ − 1
s(1)sin
(s(1) arcsin
(1
2
√1
αβ
))
− cos
(s(1) arcsin
(1
2
√1
αβ
))]. (E.20)
These two equations correspond to Eq. (3.116) and Eq. (3.117) which are repeatedly used in thederivation of the transcendental equation for systems of type 1, type 2 and type 3.
173
Appendix F
Three-body scattering amplitudeequations
The very long expressions of the three-body scattering amplitudes corresponding to the Feynmandiagrams shown in appendix C are summarized here in order to keep the main part of this workmore legible. In the respective parts of section 3.3 the according equation numbers are referenced.
174
Three-body scattering amplitudes where all terms with the same momentum structure are combined and where termswhich only differ by a ”primed“ or ”unprimed“ final state dimer are added together. Furthermore, the dq0 integration isalready performed using the residue theorem as explained in section 3.3.
i
(t12)γσαβ(t13)γσαβ(t23)γσαβ(t′12)γσαβ(t′13)γσαβ(t′23)γσαβ
(E,k,p) =
(−i)(
1− δP1P2
2
)S123
1
E − k2
2m3− p2
2m1− (k+p)2
2m2+ iε
g12(O12)α,σρ
g∗12(O†12)γ,ρβg∗13(O†13)γ,βρg∗23(O†23)γ,βρg′∗12(O′†12)γ,ρβg′∗13(O′†13)γ,βρg′∗23(O′†23)γ,βρ
δ(12)P1P3
δP1P2
1
δ(12)P1P3
δP1P2
1
f(3)(12)(12)
f(1)(12)(13)
f(1)(12)(23)
f(3)(12)(12′)
f(1)(12)(13′)
f(1)(12)(23′)
+ (−i)(
1− δP1P2
2
)S213
1
E − k2
2m3− p2
2m2− (k+p)2
2m1+ iε
g12(O12)α,ρσ
g∗12(O†12)γ,βρg∗13(O†13)γ,βρg∗23(O†23)γ,βρg′∗12(O′†12)γ,βρg′∗13(O′†13)γ,βρg′∗23(O′†23)γ,βρ
δ(12)P2P3
1δP1P2
δ(12)P2P3
1δP1P2
f(3)(12)(12)
f(2)(12)(13)
f(2)(12)(23)
f(3)(12)(12′)
f(2)(12)(13′)
f(2)(12)(23′)
+
(1− δP1P2
2
)S123
∫d3q
(2π)3i (t12)µναβ (E,k,q)
D12
(E − q2
2m3,q)
E − q2
2m3− p2
2m1− (p+q)2
2m2+ iε
g12(O12)µ,σρ
g∗12(O†12)γ,ρνg∗13(O†13)γ,νρg∗23(O†23)γ,νρg′∗12(O′†12)γ,ρνg′∗13(O′†13)γ,νρg′∗23(O′†23)γ,νρ
δ(12)P1P3
δP1P2
1
δ(12)P1P3
δP1P2
1
f(3)(12)(12)
f(1)(12)(13)
f(1)(12)(23)
f(3)(12)(12′)
f(1)(12)(13′)
f(1)(12)(23′)
175
+
(1− δP1P2
2
)S213
∫d3q
(2π)3i (t12)µναβ (E,k,q)
D12
(E − q2
2m3,q)
E − q2
2m3− p2
2m2− (p+q)2
2m1+ iε
g12(O12)µ,ρσ
g∗12(O†12)γ,νρg∗13(O†13)γ,νρg∗23(O†23)γ,νρg′∗12(O′†12)γ,νρg′∗13(O′†13)γ,νρg′∗23(O′†23)γ,νρ
δ(12)P2P3
1δP1P2
δ(12)P2P3
1δP1P2
f(3)(12)(12)
f(2)(12)(13)
f(2)(12)(23)
f(3)(12)(12′)
f(2)(12)(13′)
f(2)(12)(23′)
+
(1− δP1P3
2
)S132
∫d3q
(2π)3i (t13)µναβ (E,k,q)
D13
(E − q2
2m2,q)
E − q2
2m2− p2
2m1− (p+q)2
2m3+ iε
g13(O13)µ,σρ
g∗12(O†12)γ,νρg∗13(O†13)γ,ρνg∗23(O†23)γ,ρνg′∗12(O′†12)γ,νρg′∗13(O′†13)γ,ρνg′∗23(O′†23)γ,ρν
δP1P3
δ(13)P1P2
1δP1P3
δ(13)P1P2
1
f(3)(13)(12)
f(1)(13)(13)
f(1)(13)(23)
f(3)(13)(12′)
f(1)(13)(13′)
f(1)(13)(23′)
+
(1− δP1P3
2
)S312
∫d3q
(2π)3i (t13)µναβ (E,k,q)
D13
(E − q2
2m2,q)
E − q2
2m2− p2
2m3− (p+q)2
2m1+ iε
g13(O13)µ,ρσ
g∗12(O†12)γ,νρg∗13(O†13)γ,νρg∗23(O†23)γ,ρνg′∗12(O′†12)γ,νρg′∗13(O′†13)γ,νρg′∗23(O′†23)γ,ρν
1
δ(13)P2P3
δP1P3
1
δ(13)P2P3
δP1P3
f(3)(13)(12)
f(3)(13)(13)
f(3)(13)(23)
f(3)(13)(12′)
f(3)(13)(13′)
f(3)(13)(23′)
+
(1− δP2P3
2
)S231
∫d3q
(2π)3i (t23)µναβ (E,k,q)
D23
(E − q2
2m1,q)
E − q2
2m1− p2
2m2− (p+q)2
2m3+ iε
g23(O23)µ,σρ
g∗12(O†12)γ,ρνg∗13(O†13)γ,ρνg∗23(O†23)γ,ρνg′∗12(O′†12)γ,ρνg′∗13(O′†13)γ,ρνg′∗23(O′†23)γ,ρν
δP2P3
1
δ(23)P1P2
δP2P3
1
δ(23)P1P2
f(3)(23)(12)
f(2)(23)(13)
f(2)(23)(23)
f(3)(23)(12′)
f(2)(23)(13′)
f(2)(23)(23′)
176
+
(1− δP2P3
2
)S321
∫d3q
(2π)3i (t23)µναβ (E,k,q)
D23
(E − q2
2m1,q)
E − q2
2m1− p2
2m3− (p+q)2
2m2+ iε
g23(O23)µ,ρσ
g∗12(O†12)γ,ρνg∗13(O†13)γ,ρνg∗23(O†23)γ,νρg′∗12(O′†12)γ,ρνg′∗13(O′†13)γ,ρνg′∗23(O′†23)γ,νρ
1δP2P3
δ(23)P1P3
1δP2P3
δ(23)P1P3
f(3)(23)(12)
f(3)(23)(13)
f(3)(23)(23)
f(3)(23)(12′)
f(3)(23)(13′)
f(3)(23)(23′)
+
(1− δP1P2
2
)S123
∫d3q
(2π)3i (t′12)
µναβ (E,k,q)
D′12
(E − q2
2m3,q)
E − q2
2m3− p2
2m1− (p+q)2
2m2+ iε
g′12(O′12)µ,σρ
g∗12(O†12)γ,ρνg∗13(O†13)γ,νρg∗23(O†23)γ,νρg′∗12(O′†12)γ,ρνg′∗13(O′†13)γ,νρg′∗23(O′†23)γ,νρ
δ(12′)P1P3
δP1P2
1
δ(12′)P1P3
δP1P2
1
f(3)(12′)(12)
f(1)(12′)(13)
f(1)(12′)(23)
f(3)(12′)(23′)
f(1)(12′)(13′)
f(1)(12′)(23′)
+
(1− δP1P2
2
)S213
∫d3q
(2π)3i (t′12)
µναβ (E,k,q)
D′12
(E − q2
2m3,q)
E − q2
2m3− p2
2m2− (p+q)2
2m1+ iε
g′12(O′12)µ,ρσ
g∗12(O†12)γ,νρg∗13(O†13)γ,νρg∗23(O†23)γ,νρg′∗12(O′†12)γ,νρg′∗13(O′†13)γ,νρg′∗23(O′†23)γ,νρ
δ(12′)P2P3
1δP1P2
δ(12′)P2P3
1δP1P2
f(3)(12′)(12)
f(2)(12′)(13)
f(2)(12′)(23)
f(3)(12′)(12′)
f(2)(12′)(13′)
f(2)(12′)(23′)
+
(1− δP1P3
2
)S132
∫d3q
(2π)3i (t′13)
µναβ (E,k,q)
D′13
(E − q2
2m2,q)
E − q2
2m2− p2
2m1− (p+q)2
2m3+ iε
g′13(O′13)µ,σρ
g∗12(O†12)γ,νρg∗13(O†13)γ,ρνg∗23(O†23)γ,ρνg′∗12(O′†12)γ,νρg′∗13(O′†13)γ,ρνg′∗23(O′†23)γ,ρν
δP1P3
δ(13′)P1P2
1δP1P3
δ(13′)P1P2
1
f(3)(13′)(12)
f(1)(13′)(13)
f(1)(13′)(23)
f(3)(13′)(12′)
f(1)(13′)(13′)
f(1)(13′)(23′)
177
+
(1− δP1P3
2
)S312
∫d3q
(2π)3i (t′13)
µναβ (E,k,q)
D′13
(E − q2
2m2,q)
E − q2
2m2− p2
2m3− (p+q)2
2m1+ iε
g′13(O′13)µ,ρσ
g∗12(O†12)γ,νρg∗13(O†13)γ,νρg∗23(O†23)γ,ρνg′∗12(O′†12)γ,νρg′∗13(O′†13)γ,νρg′∗23(O′†23)γ,ρν
1
δ(13′)P2P3
δP1P3
1
δ(13′)P2P3
δP1P3
f(3)(13′)(12)
f(3)(13′)(13)
f(3)(13′)(23)
f(3)(13′)(12′)
f(3)(13′)(13′)
f(3)(13′)(23′)
+
(1− δP2P3
2
)S231
∫d3q
(2π)3i (t′23)
µναβ (E,k,q)
D′23
(E − q2
2m1,q)
E − q2
2m1− p2
2m2− (p+q)2
2m3+ iε
g′23(O′23)µ,σρ
g∗12(O†12)γ,ρνg∗13(O†13)γ,ρνg∗23(O†23)γ,ρνg′∗12(O′†12)γ,ρνg′∗13(O′†13)γ,ρνg′∗23(O′†23)γ,ρν
δP2P3
1
δ(23′)P1P2
δP2P3
1
δ(23′)P1P2
f(3)(23′)(12)
f(2)(23′)(13)
f(2)(23′)(23)
f(3)(23′)(12′)
f(2)(23′)(13′)
f(2)(23′)(23′)
+
(1− δP2P3
2
)S321
∫d3q
(2π)3i (t′23)
µναβ (E,k,q)
D′23
(E − q2
2m1,q)
E − q2
2m1− p2
2m3− (p+q)2
2m2+ iε
g′23(O′23)µ,ρσ
g∗12(O†12)γ,ρνg∗13(O†13)γ,ρνg∗23(O†23)γ,νρg′∗12(O′†12)γ,ρνg′∗13(O′†13)γ,ρνg′∗23(O′†23)γ,νρ
1δP2P3
δ(23′)P1P3
1δP2P3
δ(23′)P1P3
f(3)(23′)(12)
f(3)(23′)(13)
f(3)(23′)(23)
f(3)(23′)(12′)
f(3)(23′)(13′)
f(3)(23′)(23′)
(F.1)
178
Three-body scattering amplitudes after applying wave function renormalization (see section 3.3.1). Additionally, thefull dimer propagators are plugged in explicitly.
(T12)γσαβ(T13)γσαβ(T23)γσαβ(T ′12)γσαβ(T ′13)γσαβ(T ′23)γσαβ
(E,k,p) =
−(
1− δP1P2
2
)
(µ212 S12 c12)
−1
(µ13 µ12
√S13 S12 c13 c12
)−1
(µ23 µ12
√S23 S12 c23 c12
)−1
(µ2
12 S12
√c′12 c12
)−1
(µ13 µ12
√S13 S12 c′13 c12
)−1
(µ23 µ12
√S23 S12 c′23 c12
)−1
S123 2π γ12 (O12)α,σρ
E − k2
2m3− p2
2m1− (k+p)2
2m2+ iε
(O†12)γ,ρβ(O†13)γ,βρ(O†23)γ,βρ(O′†12)γ,ρβ(O′†13)γ,βρ(O′†23)γ,βρ
δ(12)P1P3
δP1P2
1
δ(12)P1P3
δP1P2
1
f(3)(12)(12)
f(1)(12)(13)
f(1)(12)(23)
f(3)(12)(12′)
f(1)(12)(13′)
f(1)(12)(23′)
−(
1− δP1P2
2
)
(µ212 S12 c12)
−1
(µ13 µ12
√S13S12 c13c12
)−1
(µ23 µ12
√S23S12 c23c12
)−1
(µ2
12S12
√c′12c12
)−1
(µ13 µ12
√S13S12 c′13c12
)−1
(µ23 µ12
√S23S12 c′23c12
)−1
S213 2π γ12 (O12)α,ρσ
E − k2
2m3− p2
2m2− (k+p)2
2m1+ iε
(O†12)γ,βρ(O†13)γ,βρ(O†23)γ,βρ(O′†12)γ,βρ(O′†13)γ,βρ(O′†23)γ,βρ
δ(12)P2P3
1δP1P2
δ(12)P2P3
1δP1P2
f(3)(12)(12)
f(2)(12)(13)
f(2)(12)(23)
f(3)(12)(12′)
f(2)(12)(13′)
f(2)(12)(23′)
179
−(
1− δP1P2
2
)S123 2π
(µ12 S12 c12)−1
(µ13
√S13 S12 c13 c12
)−1
(µ23
√S23 S12 c23 c12
)−1
(µ12 S12
√c′12 c12
)−1
(µ13
√S13 S12 c′13 c12
)−1
(µ23
√S23 S12 c′23 c12
)−1
(O12)µ,σρ
(O†12)γ,ρν(O†13)γ,νρ(O†23)γ,νρ(O′†12)γ,ρν(O′†13)γ,νρ(O′†23)γ,νρ
δ(12)P1P3
δP1P2
1
δ(12)P1P3
δP1P2
1
f(3)(12)(12)
f(1)(12)(13)
f(1)(12)(23)
f(3)(12)(12′)
f(1)(12)(13′)
f(1)(12)(23′)
×∫
d3q
(2π)3
(T12)µναβ (E,k,q)
−γ12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
1
E − q2
2m3− p2
2m1− (p+q)2
2m2+ iε
−(
1− δP1P2
2
)S213 2π
(µ12 S12 c12)−1
(µ13
√S13 S12 c13 c12
)−1
(µ23
√S23 S12 c23 c12
)−1
(µ12 S12
√c′12 c12
)−1
(µ13
√S13 S12 c′13 c12
)−1
(µ23
√S23 S12 c′23 c12
)−1
(O12)µ,ρσ
(O†12)γ,νρ(O†13)γ,νρ(O†23)γ,νρ(O′†12)γ,νρ(O′†13)γ,νρ(O′†23)γ,νρ
δ(12)P2P3
1δP1P2
δ(12)P2P3
1δP1P2
f(3)(12)(12)
f(2)(12)(13)
f(2)(12)(23)
f(3)(12)(12′)
f(2)(12)(13′)
f(2)(12)(23′)
×∫
d3q
(2π)3
(T12)µναβ (E,k,q)
−γ12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
1
E − q2
2m3− p2
2m2− (p+q)2
2m1+ iε
180
−(
1− δP1P3
2
)S132 2π
(µ12
√S12 S13 c12 c13
)−1
(µ13 S13 c13)−1
(µ23
√S23 S13 c23 c13
)−1
(µ12
√S12 S13 c′12 c13
)−1
(µ13 S13
√c′13 c13
)−1
(µ23
√S23 S13 c′23 c13
)−1
(O13)µ,σρ
(O†12)γ,νρ(O†13)γ,ρν(O†23)γ,ρν(O′†12)γ,νρ(O′†13)γ,ρν(O′†23)γ,ρν
δP1P3
δ(13)P1P2
1δP1P3
δ(13)P1P2
1
f(3)(13)(12)
f(1)(13)(13)
f(1)(13)(23)
f(3)(13)(12′)
f(1)(13)(13′)
f(1)(13)(23′)
×∫
d3q
(2π)3
(T13)µναβ (E,k,q)
−γ13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
1
E − q2
2m2− p2
2m1− (p+q)2
2m3+ iε
−(
1− δP1P3
2
)S312 2π
(µ12
√S12 S13 c12 c13
)−1
(µ13 S13 c13)−1
(µ23
√S23 S13 c23 c13
)−1
(µ12
√S12 S13 c′12 c13
)−1
(µ13 S13
√c′13 c13
)−1
(µ23
√S23 S13 c′23 c13
)−1
(O13)µ,ρσ
(O†12)γ,νρ(O†13)γ,νρ(O†23)γ,ρν(O′†12)γ,νρ(O′†13)γ,νρ(O′†23)γ,ρν
1
δ(13)P2P3
δP1P3
1
δ(13)P2P3
δP1P3
f(3)(13)(12)
f(3)(13)(13)
f(3)(13)(23)
f(3)(13)(12′)
f(3)(13)(13′)
f(3)(13)(23′)
×∫
d3q
(2π)3
(T13)µναβ (E,k,q)
−γ13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
1
E − q2
2m2− p2
2m3− (p+q)2
2m1+ iε
181
−(
1− δP2P3
2
)S231 2π
(µ12
√S12 S23 c12 c23
)−1
(µ13
√S13 S23 c13 c23
)−1
(µ23 S23 c23)−1
(µ12
√S12 S23 c′12 c23
)−1
(µ13
√S13 S23 c′13 c23
)−1
(µ23 S23
√c′23 c23
)−1
(O23)µ,σρ
(O†12)γ,ρν(O†13)γ,ρν(O†23)γ,ρν(O′†12)γ,ρν(O′†13)γ,ρν(O′†23)γ,ρν
δP2P3
1
δ(23)P1P2
δP2P3
1
δ(23)P1P2
f(3)(23)(12)
f(2)(23)(13)
f(2)(23)(23)
f(3)(23)(12′)
f(2)(23)(13′)
f(2)(23)(23′)
×∫
d3q
(2π)3
(T23)µναβ (E,k,q)
−γ23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
1
E − q2
2m1− p2
2m2− (p+q)2
2m3+ iε
−(
1− δP2P3
2
)S321 2π
(µ12
√S12 S23 c12 c23
)−1
(µ13
√S13 S23 c13 c23
)−1
(µ23 S23 c23)−1
(µ12
√S12 S23 c′12 c23
)−1
(µ13
√S13 S23 c′13 c23
)−1
(µ23 S23
√c′23 c23
)−1
(O23)µ,ρσ
(O†12)γ,ρν(O†13)γ,ρν(O†23)γ,νρ(O′†12)γ,ρν(O′†13)γ,ρν(O′†23)γ,νρ
1δP2P3
δ(23)P1P3
1δP2P3
δ(23)P1P3
f(3)(23)(12)
f(3)(23)(13)
f(3)(23)(23)
f(3)(23)(12′)
f(3)(23)(13′)
f(3)(23)(23′)
×∫
d3q
(2π)3
(T23)µναβ (E,k,q)
−γ23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
1
E − q2
2m1− p2
2m3− (p+q)2
2m2+ iε
182
−(
1− δP1P2
2
)S123 2π
(µ12 S12
√c12 c′12
)−1
(µ13
√S13 S12 c13 c′12
)−1
(µ23
√S23 S12 c23 c′12
)−1
(µ12 S12 c′12)−1
(µ13
√S13 S12 c′13 c
′12
)−1
(µ23
√S23 S12 c′23 c
′12
)−1
(O′12)µ,σρ
(O†12)γ,ρν(O†13)γ,νρ(O†23)γ,νρ(O′†12)γ,ρν(O′†13)γ,νρ(O′†23)γ,νρ
δ(12′)P1P3
δP1P2
1
δ(12′)P1P3
δP1P2
1
f(3)(12′)(12)
f(1)(12′)(13)
f(1)(12′)(23)
f(3)(12′)(23′)
f(1)(12′)(13′)
f(1)(12′)(23′)
×∫
d3q
(2π)3
(T ′12)µναβ (E,k,q)
−γ′12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
1
E − q2
2m3− p2
2m1− (p+q)2
2m2+ iε
−(
1− δP1P2
2
)S213 2π
(µ12 S12
√c12 c′12
)−1
(µ13
√S13 S12 c13 c′12
)−1
(µ23
√S23 S12 c23 c′12
)−1
(µ12 S12 c′12)−1
(µ13
√S13 S12 c′13 c
′12
)−1
(µ23
√S23 S12 c′23 c
′12
)−1
(O′12)µ,ρσ
(O†12)γ,νρ(O†13)γ,νρ(O†23)γ,νρ(O′†12)γ,νρ(O′†13)γ,νρ(O′†23)γ,νρ
δ(12′)P2P3
1δP1P2
δ(12′)P2P3
1δP1P2
f(3)(12′)(12)
f(2)(12′)(13)
f(2)(12′)(23)
f(3)(12′)(12′)
f(2)(12′)(13′)
f(2)(12′)(23′)
×∫
d3q
(2π)3
(T ′12)µναβ (E,k,q)
−γ′12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
1
E − q2
2m3− p2
2m2− (p+q)2
2m1+ iε
183
−(
1− δP1P3
2
)S132 2π
(µ12
√S12 S13 c12 c′13
)−1
(µ13 S13
√c13 c′13
)−1
(µ23
√S23 S13 c23 c′13
)−1
(µ12
√S12 S13 c′12 c
′13
)−1
(µ13 S13 c′13)−1
(µ23
√S23 S13 c′23 c
′13
)−1
(O′13)µ,σρ
(O†12)γ,νρ(O†13)γ,ρν(O†23)γ,ρν(O′†12)γ,νρ(O′†13)γ,ρν(O′†23)γ,ρν
δP1P3
δ(13′)P1P2
1δP1P3
δ(13′)P1P2
1
f(3)(13′)(12)
f(1)(13′)(13)
f(1)(13′)(23)
f(3)(13′)(12′)
f(1)(13′)(13′)
f(1)(13′)(23′)
×∫
d3q
(2π)3
(T ′13)µναβ (E,k,q)
−γ′13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
1
E − q2
2m2− p2
2m1− (p+q)2
2m3+ iε
−(
1− δP1P3
2
)S312 2π
(µ12
√S12 S13 c12 c′13
)−1
(µ13 S13
√c13 c′13
)−1
(µ23
√S23 S13 c23 c′13
)−1
(µ12
√S12 S13 c′12 c
′13
)−1
(µ13 S13 c′13)−1
(µ23
√S23 S13 c′23 c
′13
)−1
(O′13)µ,ρσ
(O†12)γ,νρ(O†13)γ,νρ(O†23)γ,ρν(O′†12)γ,νρ(O′†13)γ,νρ(O′†23)γ,ρν
1
δ(13′)P2P3
δP1P3
1
δ(13′)P2P3
δP1P3
f(3)(13′)(12)
f(3)(13′)(13)
f(3)(13′)(23)
f(3)(13′)(12′)
f(3)(13′)(13′)
f(3)(13′)(23′)
×∫
d3q
(2π)3
(T ′13)µναβ (E,k,q)
−γ′13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
1
E − q2
2m2− p2
2m3− (p+q)2
2m1+ iε
184
−(
1− δP2P3
2
)S231 2π
(µ12
√S12 S23 c12 c′23
)−1
(µ13
√S13 S23 c13 c′23
)−1
(µ23 S23
√c23 c′23
)−1
(µ12
√S12 S23 c′12 c
′23
)−1
(µ13
√S13 S23 c′13 c
′23
)−1
(µ23 S23 c′23)−1
(O′23)µ,σρ
(O†12)γ,ρν(O†13)γ,ρν(O†23)γ,ρν(O′†12)γ,ρν(O′†13)γ,ρν(O′†23)γ,ρν
δP2P3
1
δ(23′)P1P2
δP2P3
1
δ(23′)P1P2
f(3)(23′)(12)
f(2)(23′)(13)
f(2)(23′)(23)
f(3)(23′)(12′)
f(2)(23′)(13′)
f(2)(23′)(23′)
×∫
d3q
(2π)3
(T ′23)µναβ (E,k,q)
−γ′23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
1
E − q2
2m1− p2
2m2− (p+q)2
2m3+ iε
−(
1− δP2P3
2
)S321 2π
(µ12
√S12 S23 c12 c′23
)−1
(µ13
√S13 S23 c13 c′23
)−1
(µ23 S23
√c23 c′23
)−1
(µ12
√S12 S23 c′12 c
′23
)−1
(µ13
√S13 S23 c′13 c
′23
)−1
(µ23 S23 c′23)−1
(O′23)µ,ρσ
(O†12)γ,ρν(O†13)γ,ρν(O†23)γ,νρ(O′†12)γ,ρν(O′†13)γ,ρν(O′†23)γ,νρ
1δP2P3
δ(23′)P1P3
1δP2P3
δ(23′)P1P3
f(3)(23′)(12)
f(3)(23′)(13)
f(3)(23′)(23)
f(3)(23′)(12′)
f(3)(23′)(13′)
f(3)(23′)(23′)
×∫
d3q
(2π)3
(T ′23)µναβ (E,k,q)
−γ′23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
1
E − q2
2m1− p2
2m3− (p+q)2
2m2+ iε
(F.2)
185
Three-body scattering amplitudes after projection onto the L-th partial wave. For details (especially on the notationQijkL ) see section 3.3.2 and appendix D.
(T
(L)12
)γσαβ(
T(L)13
)γσαβ(
T(L)23
)γσαβ(
T′(L)12
)γσαβ(
T′(L)13
)γσαβ(
T′(L)23
)γσαβ
(E, k, p) =
(−1)L(
1− δP1P2
2
)S123 2π γ12
(µ212 S12 c12)
−1
(µ13 µ12
√S13 S12 c13 c12
)−1
(µ23 µ12
√S23 S12 c23 c12
)−1
(µ2
12 S12
√c′12 c12
)−1
(µ13 µ12
√S13 S12 c′13 c12
)−1
(µ23 µ12
√S23 S12 c′23 c12
)−1
(O12)α,σρ
(O†12)γ,ρβ(O†13)γ,βρ(O†23)γ,βρ(O′†12)γ,ρβ(O′†13)γ,βρ(O′†23)γ,βρ
δ(12)P1P3
δP1P2
1
δ(12)P1P3
δP1P2
1
f(3)(12)(12)
f(1)(12)(13)
f(1)(12)(23)
f(3)(12)(12′)
f(1)(12)(13′)
f(1)(12)(23′)
m2
kpQ231L (k, p;E)
+ (−1)L(
1− δP1P2
2
)S213 2π γ12
(µ212 S12 c12)
−1
(µ13 µ12
√S13S12 c13c12
)−1
(µ23 µ12
√S23S12 c23c12
)−1
(µ2
12S12
√c′12c12
)−1
(µ13 µ12
√S13S12 c′13c12
)−1
(µ23 µ12
√S23S12 c′23c12
)−1
(O12)α,ρσ
(O†12)γ,βρ(O†13)γ,βρ(O†23)γ,βρ(O′†12)γ,βρ(O′†13)γ,βρ(O′†23)γ,βρ
δ(12)P2P3
1δP1P2
δ(12)P2P3
1δP1P2
f(3)(12)(12)
f(2)(12)(13)
f(2)(12)(23)
f(3)(12)(12′)
f(2)(12)(13′)
f(2)(12)(23′)
m1
kpQ132L (k, p;E)
186
+ (−1)L(
1− δP1P2
2
)S123
1
π
(µ12 S12 c12)−1
(µ13
√S13 S12 c13 c12
)−1
(µ23
√S23 S12 c23 c12
)−1
(µ12 S12
√c′12 c12
)−1
(µ13
√S13 S12 c′13 c12
)−1
(µ23
√S23 S12 c′23 c12
)−1
(O12)µ,σρ
(O†12)γ,ρν(O†13)γ,νρ(O†23)γ,νρ(O′†12)γ,ρν(O′†13)γ,νρ(O′†23)γ,νρ
δ(12)P1P3
δP1P2
1
δ(12)P1P3
δP1P2
1
f(3)(12)(12)
f(1)(12)(13)
f(1)(12)(23)
f(3)(12)(12′)
f(1)(12)(13′)
f(1)(12)(23′)
×∫ ∞
0
dqq2(T
(L)12
)µναβ
(E, k, q)
−γ12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
m2
qpQ231L (q, p;E)
+ (−1)L(
1− δP1P2
2
)S213
1
π
(µ12 S12 c12)−1
(µ13
√S13 S12 c13 c12
)−1
(µ23
√S23 S12 c23 c12
)−1
(µ12 S12
√c′12 c12
)−1
(µ13
√S13 S12 c′13 c12
)−1
(µ23
√S23 S12 c′23 c12
)−1
(O12)µ,ρσ
(O†12)γ,νρ(O†13)γ,νρ(O†23)γ,νρ(O′†12)γ,νρ(O′†13)γ,νρ(O′†23)γ,νρ
δ(12)P2P3
1δP1P2
δ(12)P2P3
1δP1P2
f(3)(12)(12)
f(2)(12)(13)
f(2)(12)(23)
f(3)(12)(12′)
f(2)(12)(13′)
f(2)(12)(23′)
×∫ ∞
0
dqq2(T
(L)12
)µναβ
(E, k, q)
−γ12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
m1
qpQ132L (q, p;E)
187
+ (−1)L(
1− δP1P3
2
)S132
1
π
(µ12
√S12 S13 c12 c13
)−1
(µ13 S13 c13)−1
(µ23
√S23 S13 c23 c13
)−1
(µ12
√S12 S13 c′12 c13
)−1
(µ13 S13
√c′13 c13
)−1
(µ23
√S23 S13 c′23 c13
)−1
(O13)µ,σρ
(O†12)γ,νρ(O†13)γ,ρν(O†23)γ,ρν(O′†12)γ,νρ(O′†13)γ,ρν(O′†23)γ,ρν
δP1P3
δ(13)P1P2
1δP1P3
δ(13)P1P2
1
f(3)(13)(12)
f(1)(13)(13)
f(1)(13)(23)
f(3)(13)(12′)
f(1)(13)(13′)
f(1)(13)(23′)
×∫ ∞
0
dqq2(T
(L)13
)µναβ
(E, k, q)
−γ13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
m3
qpQ321L (q, p;E)
+ (−1)L(
1− δP1P3
2
)S312
1
π
(µ12
√S12 S13 c12 c13
)−1
(µ13 S13 c13)−1
(µ23
√S23 S13 c23 c13
)−1
(µ12
√S12 S13 c′12 c13
)−1
(µ13 S13
√c′13 c13
)−1
(µ23
√S23 S13 c′23 c13
)−1
(O13)µ,ρσ
(O†12)γ,νρ(O†13)γ,νρ(O†23)γ,ρν(O′†12)γ,νρ(O′†13)γ,νρ(O′†23)γ,ρν
1
δ(13)P2P3
δP1P3
1
δ(13)P2P3
δP1P3
f(3)(13)(12)
f(3)(13)(13)
f(3)(13)(23)
f(3)(13)(12′)
f(3)(13)(13′)
f(3)(13)(23′)
×∫ ∞
0
dqq2(T
(L)13
)µναβ
(E, k, q)
−γ13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
m1
qpQ123L (q, p;E)
188
+ (−1)L(
1− δP2P3
2
)S231
1
π
(µ12
√S12 S23 c12 c23
)−1
(µ13
√S13 S23 c13 c23
)−1
(µ23 S23 c23)−1
(µ12
√S12 S23 c′12 c23
)−1
(µ13
√S13 S23 c′13 c23
)−1
(µ23 S23
√c′23 c23
)−1
(O23)µ,σρ
(O†12)γ,ρν(O†13)γ,ρν(O†23)γ,ρν(O′†12)γ,ρν(O′†13)γ,ρν(O′†23)γ,ρν
δP2P3
1
δ(23)P1P2
δP2P3
1
δ(23)P1P2
f(3)(23)(12)
f(2)(23)(13)
f(2)(23)(23)
f(3)(23)(12′)
f(2)(23)(13′)
f(2)(23)(23′)
×∫ ∞
0
dqq2(T
(L)23
)µναβ
(E, k, q)
−γ23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
m3
qpQ312L (q, p;E)
+ (−1)L(
1− δP2P3
2
)S321
1
π
(µ12
√S12 S23 c12 c23
)−1
(µ13
√S13 S23 c13 c23
)−1
(µ23 S23 c23)−1
(µ12
√S12 S23 c′12 c23
)−1
(µ13
√S13 S23 c′13 c23
)−1
(µ23 S23
√c′23 c23
)−1
(O23)µ,ρσ
(O†12)γ,ρν(O†13)γ,ρν(O†23)γ,νρ(O′†12)γ,ρν(O′†13)γ,ρν(O′†23)γ,νρ
1δP2P3
δ(23)P1P3
1δP2P3
δ(23)P1P3
f(3)(23)(12)
f(3)(23)(13)
f(3)(23)(23)
f(3)(23)(12′)
f(3)(23)(13′)
f(3)(23)(23′)
×∫ ∞
0
dqq2(T
(L)23
)µναβ
(E, k, q)
−γ23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
m2
qpQ213L (q, p;E)
189
+ (−1)L(
1− δP1P2
2
)S123
1
π
(µ12 S12
√c12 c′12
)−1
(µ13
√S13 S12 c13 c′12
)−1
(µ23
√S23 S12 c23 c′12
)−1
(µ12 S12 c′12)−1
(µ13
√S13 S12 c′13 c
′12
)−1
(µ23
√S23 S12 c′23 c
′12
)−1
(O′12)µ,σρ
(O†12)γ,ρν(O†13)γ,νρ(O†23)γ,νρ(O′†12)γ,ρν(O′†13)γ,νρ(O′†23)γ,νρ
δ(12′)P1P3
δP1P2
1
δ(12′)P1P3
δP1P2
1
f(3)(12′)(12)
f(1)(12′)(13)
f(1)(12′)(23)
f(3)(12′)(23′)
f(1)(12′)(13′)
f(1)(12′)(23′)
×∫ ∞
0
dqq2(T′(L)12
)µναβ
(E, k, q)
−γ′12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
m2
qpQ231L (q, p;E)
+ (−1)L(
1− δP1P2
2
)S213
1
π
(µ12 S12
√c12 c′12
)−1
(µ13
√S13 S12 c13 c′12
)−1
(µ23
√S23 S12 c23 c′12
)−1
(µ12 S12 c′12)−1
(µ13
√S13 S12 c′13 c
′12
)−1
(µ23
√S23 S12 c′23 c
′12
)−1
(O′12)µ,ρσ
(O†12)γ,νρ(O†13)γ,νρ(O†23)γ,νρ(O′†12)γ,νρ(O′†13)γ,νρ(O′†23)γ,νρ
δ(12′)P2P3
1δP1P2
δ(12′)P2P3
1δP1P2
f(3)(12′)(12)
f(2)(12′)(13)
f(2)(12′)(23)
f(3)(12′)(12′)
f(2)(12′)(13′)
f(2)(12′)(23′)
×∫ ∞
0
dqq2(T′(L)12
)µναβ
(E, k, q)
−γ′12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
m1
qpQ132L (q, p;E)
190
+ (−1)L(
1− δP1P3
2
)S132
1
π
(µ12
√S12 S13 c12 c′13
)−1
(µ13 S13
√c13 c′13
)−1
(µ23
√S23 S13 c23 c′13
)−1
(µ12
√S12 S13 c′12 c
′13
)−1
(µ13 S13 c′13)−1
(µ23
√S23 S13 c′23 c
′13
)−1
(O′13)µ,σρ
(O†12)γ,νρ(O†13)γ,ρν(O†23)γ,ρν(O′†12)γ,νρ(O′†13)γ,ρν(O′†23)γ,ρν
δP1P3
δ(13′)P1P2
1δP1P3
δ(13′)P1P2
1
f(3)(13′)(12)
f(1)(13′)(13)
f(1)(13′)(23)
f(3)(13′)(12′)
f(1)(13′)(13′)
f(1)(13′)(23′)
×∫ ∞
0
dqq2(T′(L)13
)µναβ
(E, k, q)
−γ′13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
m3
qpQ321L (q, p;E)
+ (−1)L(
1− δP1P3
2
)S312
1
π
(µ12
√S12 S13 c12 c′13
)−1
(µ13 S13
√c13 c′13
)−1
(µ23
√S23 S13 c23 c′13
)−1
(µ12
√S12 S13 c′12 c
′13
)−1
(µ13 S13 c′13)−1
(µ23
√S23 S13 c′23 c
′13
)−1
(O′13)µ,ρσ
(O†12)γ,νρ(O†13)γ,νρ(O†23)γ,ρν(O′†12)γ,νρ(O′†13)γ,νρ(O′†23)γ,ρν
1
δ(13′)P2P3
δP1P3
1
δ(13′)P2P3
δP1P3
f(3)(13′)(12)
f(3)(13′)(13)
f(3)(13′)(23)
f(3)(13′)(12′)
f(3)(13′)(13′)
f(3)(13′)(23′)
×∫ ∞
0
dqq2(T′(L)13
)µναβ
(E, k, q)
−γ′13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
m1
qpQ123L (q, p;E)
191
+ (−1)L(
1− δP2P3
2
)S231
1
π
(µ12
√S12 S23 c12 c′23
)−1
(µ13
√S13 S23 c13 c′23
)−1
(µ23 S23
√c23 c′23
)−1
(µ12
√S12 S23 c′12 c
′23
)−1
(µ13
√S13 S23 c′13 c
′23
)−1
(µ23 S23 c′23)−1
(O′23)µ,σρ
(O†12)γ,ρν(O†13)γ,ρν(O†23)γ,ρν(O′†12)γ,ρν(O′†13)γ,ρν(O′†23)γ,ρν
δP2P3
1
δ(23′)P1P2
δP2P3
1
δ(23′)P1P2
f(3)(23′)(12)
f(2)(23′)(13)
f(2)(23′)(23)
f(3)(23′)(12′)
f(2)(23′)(13′)
f(2)(23′)(23′)
×∫ ∞
0
dqq2(T′(L)23
)µναβ
(E, k, q)
−γ′23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
m3
qpQ312L (q, p;E)
+ (−1)L(
1− δP2P3
2
)S321
1
π
(µ12
√S12 S23 c12 c′23
)−1
(µ13
√S13 S23 c13 c′23
)−1
(µ23 S23
√c23 c′23
)−1
(µ12
√S12 S23 c′12 c
′23
)−1
(µ13
√S13 S23 c′13 c
′23
)−1
(µ23 S23 c′23)−1
(O′23)µ,ρσ
(O†12)γ,ρν(O†13)γ,ρν(O†23)γ,νρ(O′†12)γ,ρν(O′†13)γ,ρν(O′†23)γ,νρ
1δP2P3
δ(23′)P1P3
1δP2P3
δ(23′)P1P3
f(3)(23′)(12)
f(3)(23′)(13)
f(3)(23′)(23)
f(3)(23′)(12′)
f(3)(23′)(13′)
f(3)(23′)(23′)
×∫ ∞
0
dqq2(T′(L)23
)µναβ
(E, k, q)
−γ′23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
m2
qpQ213L (q, p;E) (F.3)
Three-body scattering amplitudes after projection onto a specific isospin and spin channel as explained in section 3.3.3.For the definition of the xi, yi and zi parameters see appendix A.2.
192
T(L)12 (E, k, p) =
(−1)L(
1− δP1P2
2
)2 π γ12
µ212 S12 c12
[x1 δ
(12)P1P3
f(3)(12)(12) S123
m2
kpQ211L (k, p;E) + x1 δ
(12)P2P3
f(3)(12)(12) S213
m1
kpQ122L (k, p;E)
]
+ (−1)L(
1− δP1P2
2
)1
π
1
µ12 S12 c12
∫ ∞
0
dqq2 T
(L)12 (E, k, q)
−γ12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
×[x2 δ
(12)P1P3
f(3)(12)(12) S123
m2
qpQ211L (q, p;E) + x2 δ
(12)P2P3
f(3)(12)(12) S213
m1
qpQ122L (q, p;E)
]
+ (−1)L(
1− δP1P3
2
)1
π
(x3 δP1P3 S132 + x3 S312
)
µ12
√S12 S13 c12 c13
f(3)(13)(12)
∫ ∞
0
dqq2 T
(L)13 (E, k, q) m1
qpQ123L (q, p;E)
−γ13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
+ (−1)L(
1− δP2P3
2
)1
π
(x4 δP2P3 S231 + x4 S321
)
µ12
√S12 S23 c12 c23
f(3)(23)(12)
∫ ∞
0
dqq2 T
(L)23 (E, k, q) m2
qpQ213L (q, p;E)
−γ23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
+ (−1)L(
1− δP1P2
2
)1
π
1
µ12 S12
√c12 c′12
∫ ∞
0
dqq2 T
′(L)12 (E, k, q)
−γ′12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
×[x5 δ
(12′)P1P3
f(3)(12′)(12) S123
m2
qpQ211L (q, p;E) + x5 δ
(12′)P2P3
f(3)(12′)(12) S213
m1
qpQ122L (q, p;E)
]
+ (−1)L(
1− δP1P3
2
)1
π
(x6 δP1P3 S132 + x6 S312
)
µ12
√S12 S13 c12 c′13
f(3)(13′)(12)
∫ ∞
0
dqq2 T
′(L)13 (E, k, q) m1
qpQ123L (q, p;E)
−γ′13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
+ (−1)L(
1− δP2P3
2
)1
π
(x7 δP2P3 S231 + x7 S321
)
µ12
√S12 S23 c12 c′23
f(3)(23′)(12)
∫ ∞
0
dqq2 T
′(L)23 (E, k, q) m2
qpQ213L (q, p;E)
−γ′23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
(F.4)
193
T(L)13 (E, k, p) =
(−1)L(
1− δP1P2
2
) 2 π γ12
(y1 δP1P2 S123 + y1 S213
)
µ13 µ12
√S13 S12 c13 c12
f(2)(12)(13)
m1
kpQ132L (k, p;E)
+ (−1)L(
1− δP1P2
2
)1
π
(y2 δP1P2 S123 + y2 S213
)
µ13
√S13 S12 c13 c12
f(2)(12)(13)
∫ ∞
0
dqq2 T
(L)12 (E, k, q) m1
qpQ132L (q, p;E)
−γ12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
+ (−1)L(
1− δP1P3
2
)1
π
1
µ13 S13 c13
∫ ∞
0
dqq2 T
(L)13 (E, k, q)
−γ13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
×[y3 δ
(13)P1P2
f(1)(13)(13) S132
m3
qpQ311L (q, p;E) + y3 δ
(13)P2P3
f(3)(13)(13) S312
m1
qpQ133L (q, p;E)
]
+ (−1)L(
1− δP2P3
2
)1
π
(y4 S231 + y4 δP2P3 S321
)
µ13
√S13 S23 c13 c23
f(2)(23)(13)
∫ ∞
0
dqq2 T
(L)23 (E, k, q) m3
qpQ312L (q, p;E)
−γ23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
+ (−1)L(
1− δP1P2
2
)1
π
(y5 δP1P2 S123 + y5 S213
)
µ13
√S13 S12 c13 c′12
f(2)(12′)(13)
∫ ∞
0
dqq2 T
′(L)12 (E, k, q) m1
qpQ132L (q, p;E)
−γ′12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
+ (−1)L(
1− δP1P3
2
)1
π
1
µ13 S13
√c13 c′13
∫ ∞
0
dqq2 T
′(L)13 (E, k, q)
−γ′13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
×[y6 δ
(13′)P1P2
f(1)(13′)(13) S132
m3
qpQ311L (q, p;E) + y6 δ
(13′)P2P3
f(3)(13′)(13) S312
m1
qpQ133L (q, p;E)
]
+ (−1)L(
1− δP2P3
2
)1
π
(y7 S231 + y7 δP2P3 S321
)
µ13
√S13 S23 c13 c′23
f(2)(23′)(13)
∫ ∞
0
dqq2 T
′(L)23 (E, k, q) m3
qpQ312L (q, p;E)
−γ′23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
(F.5)
194
T(L)23 (E, k, p) =
(−1)L(
1− δP1P2
2
) 2 π γ12
(z1 S123 + z1 δP1P2 S213
)
µ23 µ12
√S23 S12 c23 c12
f(1)(12)(23)
m2
kpQ231L (k, p;E)
+ (−1)L(
1− δP1P2
2
)1
π
(z2 S123 + z2 δP1P2 S213
)
µ23
√S23 S12 c23 c12
f(1)(12)(23)
∫ ∞
0
dqq2 T
(L)12 (E, k, q) m2
qpQ231L (q, p;E)
−γ12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
+ (−1)L(
1− δP1P3
2
)1
π
(z3 S132 + z3 δP1P3 S312
)
µ23
√S23 S13 c23 c13
f(1)(13)(23)
∫ ∞
0
dqq2 T
(L)13 (E, k, q) m3
qpQ321L (q, p;E)
−γ13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
+ (−1)L(
1− δP2P3
2
)1
π
1
µ23 S23 c23
∫ ∞
0
dqq2 T
(L)23 (E, k, q)
−γ23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
×[z4 δ
(23)P1P2
f(2)(23)(23) S231
m3
qpQ322L (q, p;E) + z4 δ
(23)P1P3
f(3)(23)(23) S321
m2
qpQ233L (q, p;E)
]
+ (−1)L(
1− δP1P2
2
)1
π
(z5 S123 + z5 δP1P2 S213
)
µ23
√S23 S12 c23 c′12
f(1)(12′)(23)
∫ ∞
0
dqq2 T
′(L)12 (E, k, q) m2
qpQ231L (q, p;E)
−γ′12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
+ (−1)L(
1− δP1P3
2
)1
π
(z6 S132 + z6 δP1P3 S312
)
µ23
√S23 S13 c23 c′13
f(1)(13′)(23)
∫ ∞
0
dqq2 T
′(L)13 (E, k, q) m3
qpQ321L (q, p;E)
−γ′13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
+ (−1)L(
1− δP2P3
2
)1
π
1
µ23 S23
√c23 c′23
∫ ∞
0
dqq2 T
′(L)23 (E, k, q)
−γ′23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
×[z7 δ
(23′)P1P2
f(2)(23′)(23) S231
m3
qpQ322L (q, p;E) + z7 δ
(23′)P1P3
f(3)(23′)(23) S321
m2
qpQ233L (q, p;E)
](F.6)
195
T′(L)12 (E, k, p) =
(−1)L(
1− δP1P2
2
)2 π γ12
µ212 S12
√c′12 c12
[x′1 δ
(12)P1P3
f(3)(12)(12′) S123
m2
kpQ211L (k, p;E) + x′1 δ
(12)P2P3
f(3)(12)(12′) S213
m1
kpQ122L (k, p;E)
]
+ (−1)L(
1− δP1P2
2
)1
π
1
µ12 S12
√c′12 c12
∫ ∞
0
dqq2 T
(L)12 (E, k, q)
−γ12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
×[x′2 δ
(12)P1P3
f(3)(12)(12′) S123
m2
qpQ211L (q, p;E) + x′2 δ
(12)P2P3
f(3)(12)(12′) S213
m1
qpQ122L (q, p;E)
]
+ (−1)L(
1− δP1P3
2
)1
π
(x′3 δP1P3 S132 + x′3 S312
)
µ12
√S12 S13 c′12 c13
f(3)(13)(12′)
∫ ∞
0
dqq2 T
(L)13 (E, k, q) m1
qpQ123L (q, p;E)
−γ13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
+ (−1)L(
1− δP2P3
2
)1
π
(x′4 δP2P3 S231 + x′4 S321
)
µ12
√S12 S23 c′12 c23
f(3)(23)(12′)
∫ ∞
0
dqq2 T
(L)23 (E, k, q) m2
qpQ213L (q, p;E)
−γ23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
+ (−1)L(
1− δP1P2
2
)1
π
1
µ12 S12 c′12
∫ ∞
0
dqq2 T
′(L)12 (E, k, q)
−γ′12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
×[x′5 δ
(12′)P1P3
f(3)(12′)(23′) S123
m2
qpQ211L (q, p;E) + x′5 δ
(12′)P2P3
f(3)(12′)(12′) S213
m1
qpQ122L (q, p;E)
]
+ (−1)L(
1− δP1P3
2
)1
π
(x′6 δP1P3 S132 + x′6 S312
)
µ12
√S12 S13 c′12 c
′13
f(3)(13′)(12′)
∫ ∞
0
dqq2 T
′(L)13 (E, k, q) m1
qpQ123L (q, p;E)
−γ′13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
+ (−1)L(
1− δP2P3
2
)1
π
(x′7 δP2P3 S231 + x′7 S321
)
µ12
√S12 S23 c′12 c
′23
f(3)(23′)(12′)
∫ ∞
0
dqq2 T
′(L)23 (E, k, q) m2
qpQ213L (q, p;E)
−γ′23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
(F.7)
196
T′(L)13 (E, k, p) =
(−1)L(
1− δP1P2
2
) 2 π γ12
(y′1 δP1P2 S123 + y′1 S213
)
µ13 µ12
√S13 S12 c′13 c12
f(2)(12)(13′)
m1
kpQ132L (k, p;E)
+ (−1)L(
1− δP1P2
2
)1
π
(y′2 δP1P2 S123 + y′2 S213
)
µ13
√S13 S12 c′13 c12
f(2)(12)(13′)
∫ ∞
0
dqq2 T
(L)12 (E, k, q) m1
qpQ132L (q, p;E)
−γ12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
+ (−1)L(
1− δP1P3
2
)1
π
1
µ13 S13
√c′13 c13
∫ ∞
0
dqq2 T
(L)13 (E, k, q)
−γ13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
×[y′3 δ
(13)P1P2
f(1)(13)(13′) S132
m3
qpQ311L (q, p;E) + y′3 δ
(13)P2P3
f(3)(13)(13′) S312
m1
qpQ133L (q, p;E)
]
+ (−1)L(
1− δP2P3
2
)1
π
(y′4 S231 + y′4 δP2P3 S321
)
µ13
√S13 S23 c′13 c23
f(2)(23)(13′)
∫ ∞
0
dqq2 T
(L)23 (E, k, q) m3
qpQ312L (q, p;E)
−γ23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
+ (−1)L(
1− δP1P2
2
)1
π
(y′5 δP1P2 S123 + y′5 S213
)
µ13
√S13 S12 c′13 c
′12
f(2)(12′)(13′)
∫ ∞
0
dqq2 T
′(L)12 (E, k, q) m1
qpQ132L (q, p;E)
−γ′12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
+ (−1)L(
1− δP1P3
2
)1
π
1
µ13 S13 c′13
∫ ∞
0
dqq2 T
′(L)13 (E, k, q)
−γ′13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
×[y′6 δ
(13′)P1P2
f(1)(13′)(13′) S132
m3
qpQ311L (q, p;E) + y′6 δ
(13′)P2P3
f(3)(13′)(13′) S312
m1
qpQ133L (q, p;E)
]
+ (−1)L(
1− δP2P3
2
)1
π
(y′7 S231 + y′7 δP2P3 S321
)
µ13
√S13 S23 c′13 c
′23
f(2)(23′)(13′)
∫ ∞
0
dqq2 T
′(L)23 (E, k, q) m3
qpQ312L (q, p;E)
−γ′23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
(F.8)
197
T′(L)23 (E, k, p) =
(−1)L(
1− δP1P2
2
) 2 π γ12
(z′1 S123 + z′1 δP1P2 S213
)
µ23 µ12
√S23 S12 c′23 c12
f(1)(12)(23′)
m2
kpQ231L (k, p;E)
+ (−1)L(
1− δP1P2
2
)1
π
(z′2 S123 + z′2 δP1P2 S213
)
µ23
√S23 S12 c′23 c12
f(1)(12)(23′)
∫ ∞
0
dqq2 T
(L)12 (E, k, q) m2
qpQ231L (q, p;E)
−γ12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
+ (−1)L(
1− δP1P3
2
)1
π
(z′3 S132 + z′3 δP1P3 S312
)
µ23
√S23 S13 c′23 c13
f(1)(13)(23′)
∫ ∞
0
dqq2 T
(L)13 (E, k, q) m3
qpQ321L (q, p;E)
−γ13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
+ (−1)L(
1− δP2P3
2
)1
π
1
µ23 S23
√c′23 c23
∫ ∞
0
dqq2 T
(L)23 (E, k, q)
−γ23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
×[z′4 δ
(23)P1P2
f(2)(23)(23′) S231
m3
qpQ322L (q, p;E) + z′4 δ
(23)P1P3
f(3)(23)(23′) S321
m2
qpQ233L (q, p;E)
]
+ (−1)L(
1− δP1P2
2
)1
π
(z′5 S123 + z′5 δP1P2 S213
)
µ23
√S23 S12 c′23 c
′12
f(1)(12′)(23′)
∫ ∞
0
dqq2 T
′(L)12 (E, k, q) m2
qpQ231L (q, p;E)
−γ′12 +
√−2µ12
(E − q2
2m3− q2
2(m1+m2)
)− iε
+ (−1)L(
1− δP1P3
2
)1
π
(z′6 S132 + z′6 δP1P3 S312
)
µ23
√S23 S13 c′23 c
′13
f(1)(13′)(23′)
∫ ∞
0
dqq2 T
′(L)13 (E, k, q) m3
qpQ321L (q, p;E)
−γ′13 +
√−2µ13
(E − q2
2m2− q2
2(m1+m3)
)− iε
+ (−1)L(
1− δP2P3
2
)1
π
1
µ23 S23 c′23
∫ ∞
0
dqq2 T
′(L)23 (E, k, q)
−γ′23 +
√−2µ23
(E − q2
2m1− q2
2(m2+m3)
)− iε
×[z′7 δ
(23′)P1P2
f(2)(23′)(23′) S231
m3
qpQ322L (q, p;E) + z′7 δ
(23′)P1P3
f(3)(23′)(23′) S321
m2
qpQ233L (q, p;E)
](F.9)
198
Three-body scattering amplitudes in the limit of asymptotic large off-shell momenta (see section 3.4).
T(L)12 (p) = (−1)L
(1− δP1P2
2
)1
π
1
µ12 S12 c12
∫ ΛC
0
dq
q
T(L)12 (q)√µ12µ(12)3
×[x2 δ
(12)P1P3
f(3)(12)(12) S123 m2Q
211L (q, p) + x2 δ
(12)P2P3
f(3)(12)(12) S213 m1Q
122L (q, p)
]
+ (−1)L(
1− δP1P3
2
)1
π
(x3 δP1P3 S132 + x3 S312
)
µ12
√S12 S13 c12 c13
f(3)(13)(12)
∫ ΛC
0
dq
q
T(L)13 (q)√µ13µ(13)2
m1Q123L (q, p)
+ (−1)L(
1− δP2P3
2
)1
π
(x4 δP2P3 S231 + x4 S321
)
µ12
√S12 S23 c12 c23
f(3)(23)(12)
∫ ΛC
0
dq
q
T(L)23 (q)√µ23µ(23)1
m2Q213L (q, p)
+ (−1)L(
1− δP1P2
2
)1
π
1
µ12 S12
√c12 c′12
∫ ΛC
0
dq
q
T′(L)12 (q)√
µ12µ(12)3
×[x5 δ
(12′)P1P3
f(3)(12′)(12) S123 m2Q
211L (q, p) + x5 δ
(12′)P2P3
f(3)(12′)(12) S213 m1Q
122L (q, p)
]
+ (−1)L(
1− δP1P3
2
)1
π
(x6 δP1P3 S132 + x6 S312
)
µ12
√S12 S13 c12 c′13
f(3)(13′)(12)
∫ ΛC
0
dq
q
T′(L)13 (q)√
µ13µ(13)2
m1Q123L (q, p)
+ (−1)L(
1− δP2P3
2
)1
π
(x7 δP2P3 S231 + x7 S321
)
µ12
√S12 S23 c12 c′23
f(3)(23′)(12)
∫ ΛC
0
dq
q
T′(L)23 (q)√
µ23µ(23)1
m2Q213L (q, p) (F.10)
199
T(L)13 (p) = (−1)L
(1− δP1P2
2
)1
π
(y2 δP1P2 S123 + y2 S213
)
µ13
√S13 S12 c13 c12
f(2)(12)(13)
∫ ΛC
0
dq
q
T(L)12 (q)√µ12µ(12)3
m1Q132L (q, p)
+ (−1)L(
1− δP1P3
2
)1
π
1
µ13 S13 c13
∫ ΛC
0
dq
q
T(L)13 (q)√µ13µ(13)2
×[y3 δ
(13)P1P2
f(1)(13)(13) S132 m3Q
311L (q, p) + y3 δ
(13)P2P3
f(3)(13)(13) S312 m1Q
133L (q, p)
]
+ (−1)L(
1− δP2P3
2
)1
π
(y4 S231 + y4 δP2P3 S321
)
µ13
√S13 S23 c13 c23
f(2)(23)(13)
∫ ΛC
0
dq
q
T(L)23 (q)√µ23µ(23)1
m3Q312L (q, p)
+ (−1)L(
1− δP1P2
2
)1
π
(y5 δP1P2 S123 + y5 S213
)
µ13
√S13 S12 c13 c′12
f(2)(12′)(13)
∫ ΛC
0
dq
q
T′(L)12 (q)√
µ12µ(12)3
m1Q132L (q, p)
+ (−1)L(
1− δP1P3
2
)1
π
1
µ13 S13
√c13 c′13
∫ ΛC
0
dq
q
T′(L)13 (q)√
µ13µ(13)2
×[y6 δ
(13′)P1P2
f(1)(13′)(13) S132 m3Q
311L (q, p) + y6 δ
(13′)P2P3
f(3)(13′)(13) S312 m1Q
133L (q, p)
]
+ (−1)L(
1− δP2P3
2
)1
π
(y7 S231 + y7 δP2P3 S321
)
µ13
√S13 S23 c13 c′23
f(2)(23′)(13)
∫ ΛC
0
dq
q
T′(L)23 (q)√
µ23µ(23)1
m3Q312L (q, p) (F.11)
200
T(L)23 (p) = (−1)L
(1− δP1P2
2
)1
π
(z2 S123 + z2 δP1P2 S213
)
µ23
√S23 S12 c23 c12
f(1)(12)(23)
∫ ΛC
0
dq
q
T(L)12 (q)√µ12µ(12)3
m2Q231L (q, p)
+ (−1)L(
1− δP1P3
2
)1
π
(z3 S132 + z3 δP1P3 S312
)
µ23
√S23 S13 c23 c13
f(1)(13)(23)
∫ ΛC
0
dq
q
T(L)13 (q)√µ13µ(13)2
m3Q321L (q, p)
+ (−1)L(
1− δP2P3
2
)1
π
1
µ23 S23 c23
∫ ΛC
0
dq
q
T(L)23 (q)√µ23µ(23)1
×[z4 δ
(23)P1P2
f(2)(23)(23) S231 m3Q
322L (q, p) + z4 δ
(23)P1P3
f(3)(23)(23) S321 m2Q
233L (q, p)
]
+ (−1)L(
1− δP1P2
2
)1
π
(z5 S123 + z5 δP1P2 S213
)
µ23
√S23 S12 c23 c′12
f(1)(12′)(23)
∫ ΛC
0
dq
q
T′(L)12 (q)√
µ12µ(12)3
m2Q231L (q, p)
+ (−1)L(
1− δP1P3
2
)1
π
(z6 S132 + z6 δP1P3 S312
)
µ23
√S23 S13 c23 c′13
f(1)(13′)(23)
∫ ΛC
0
dq
q
T′(L)13 (q)√
µ13µ(13)2
m3Q321L (q, p)
+ (−1)L(
1− δP2P3
2
)1
π
1
µ23 S23
√c23 c′23
∫ ΛC
0
dq
q
T′(L)23 (q)√
µ23µ(23)1
×[z7 δ
(23′)P1P2
f(2)(23′)(23) S231 m3Q
322L (q, p) + z7 δ
(23′)P1P3
f(3)(23′)(23) S321 m2Q
233L (q, p)
](F.12)
201
T′(L)12 (p) = (−1)L
(1− δP1P2
2
)1
π
1
µ12 S12
√c′12 c12
∫ ΛC
0
dq
q
T(L)12 (q)√µ12µ(12)3
×[x′2 δ
(12)P1P3
f(3)(12)(12′) S123 m2Q
211L (q, p) + x′2 δ
(12)P2P3
f(3)(12)(12′) S213 m1Q
122L (q, p)
]
+ (−1)L(
1− δP1P3
2
)1
π
(x′3 δP1P3 S132 + x′3 S312
)
µ12
√S12 S13 c′12 c13
f(3)(13)(12′)
∫ ΛC
0
dq
q
T(L)13 (q)√µ13µ(13)2
m1Q123L (q, p)
+ (−1)L(
1− δP2P3
2
)1
π
(x′4 δP2P3 S231 + x′4 S321
)
µ12
√S12 S23 c′12 c23
f(3)(23)(12′)
∫ ΛC
0
dq
q
T(L)23 (q)√µ23µ(23)1
m2Q213L (q, p)
+ (−1)L(
1− δP1P2
2
)1
π
1
µ12 S12 c′12
∫ ΛC
0
dq
q
T′(L)12 (q)√
µ12µ(12)3
×[x′5 δ
(12′)P1P3
f(3)(12′)(23′) S123 m2Q
211L (q, p) + x′5 δ
(12′)P2P3
f(3)(12′)(12′) S213 m1Q
122L (q, p)
]
+ (−1)L(
1− δP1P3
2
)1
π
(x′6 δP1P3 S132 + x′6 S312
)
µ12
√S12 S13 c′12 c
′13
f(3)(13′)(12′)
∫ ΛC
0
dq
q
T′(L)13 (q)√
µ13µ(13)2
m1Q123L (q, p)
+ (−1)L(
1− δP2P3
2
)1
π
(x′7 δP2P3 S231 + x′7 S321
)
µ12
√S12 S23 c′12 c
′23
f(3)(23′)(12′)
∫ ΛC
0
dq
q
T′(L)23 (q)√
µ23µ(23)1
m2Q213L (q, p) (F.13)
202
T′(L)13 (p) = (−1)L
(1− δP1P2
2
)1
π
(y′2 δP1P2 S123 + y′2 S213
)
µ13
√S13 S12 c′13 c12
f(2)(12)(13′)
∫ ΛC
0
dq
q
T(L)12 (q)√µ12µ(12)3
m1Q132L (q, p)
+ (−1)L(
1− δP1P3
2
)1
π
1
µ13 S13
√c′13 c13
∫ ΛC
0
dq
q
T(L)13 (q)√µ13µ(13)2
×[y′3 δ
(13)P1P2
f(1)(13)(13′) S132 m3Q
311L (q, p) + y′3 δ
(13)P2P3
f(3)(13)(13′) S312 m1Q
133L (q, p)
]
+ (−1)L(
1− δP2P3
2
)1
π
(y′4 S231 + y′4 δP2P3 S321
)
µ13
√S13 S23 c′13 c23
f(2)(23)(13′)
∫ ΛC
0
dq
q
T(L)23 (q)√µ23µ(23)1
m3Q312L (q, p)
+ (−1)L(
1− δP1P2
2
)1
π
(y′5 δP1P2 S123 + y′5 S213
)
µ13
√S13 S12 c′13 c
′12
f(2)(12′)(13′)
∫ ΛC
0
dq
q
T′(L)12 (q)√
µ12µ(12)3
m1Q132L (q, p)
+ (−1)L(
1− δP1P3
2
)1
π
1
µ13 S13 c′13
∫ ΛC
0
dq
q
T′(L)13 (q)√
µ13µ(13)2
×[y′6 δ
(13′)P1P2
f(1)(13′)(13′) S132 m3Q
311L (q, p) + y′6 δ
(13′)P2P3
f(3)(13′)(13′) S312 m1Q
133L (q, p)
]
+ (−1)L(
1− δP2P3
2
)1
π
(y′7 S231 + y′7 δP2P3 S321
)
µ13
√S13 S23 c′13 c
′23
f(2)(23′)(13′)
∫ ΛC
0
dq
q
T′(L)23 (q)√
µ23µ(23)1
m3Q312L (q, p) (F.14)
203
T′(L)23 (p) = (−1)L
(1− δP1P2
2
)1
π
(z′2 S123 + z′2 δP1P2 S213
)
µ23
√S23 S12 c′23 c12
f(1)(12)(23′)
∫ ΛC
0
dq
q
T(L)12 (q)√µ12µ(12)3
m2Q231L (q, p)
+ (−1)L(
1− δP1P3
2
)1
π
(z′3 S132 + z′3 δP1P3 S312
)
µ23
√S23 S13 c′23 c13
f(1)(13)(23′)
∫ ΛC
0
dq
q
T(L)13 (q)√µ13µ(13)2
m3Q321L (q, p)
+ (−1)L(
1− δP2P3
2
)1
π
1
µ23 S23
√c′23 c23
∫ ΛC
0
dq
q
T(L)23 (q)√µ23µ(23)1
×[z′4 δ
(23)P1P2
f(2)(23)(23′) S231 m3Q
322L (q, p) + z′4 δ
(23)P1P3
f(3)(23)(23′) S321 m2Q
233L (q, p)
]
+ (−1)L(
1− δP1P2
2
)1
π
(z′5 S123 + z′5 δP1P2 S213
)
µ23
√S23 S12 c′23 c
′12
f(1)(12′)(23′)
∫ ΛC
0
dq
q
T′(L)12 (q)√
µ12µ(12)3
m2Q231L (q, p)
+ (−1)L(
1− δP1P3
2
)1
π
(z′6 S132 + z′6 δP1P3 S312
)
µ23
√S23 S13 c′23 c
′13
f(1)(13′)(23′)
∫ ΛC
0
dq
q
T′(L)13 (q)√
µ13µ(13)2
m3Q321L (q, p)
+ (−1)L(
1− δP2P3
2
)1
π
1
µ23 S23 c′23
∫ ΛC
0
dq
q
T′(L)23 (q)√
µ23µ(23)1
×[z′7 δ
(23′)P1P2
f(2)(23′)(23′) S231 m3Q
322L (q, p) + z′7 δ
(23′)P1P3
f(3)(23′)(23′) S321 m2Q
233L (q, p)
](F.15)
204
Appendix G
Numerical implementation of scatteringamplitudes
In section 5.1 we have derived the elastic S-wave three-body scattering amplitudes for all pos-sible Z
(′)b –B(∗) scattering processes. In this appendix it will be discussed how one can solve them
numerically in order to extract the three-body observables scattering length and phase shift.Comparing all scattering processes and their corresponding channels one concludes that all in-tegral equations have a similar structure. However, there are some differences: on the one handthey differ in the prefactors determined by the spin and isospin projections. On the other handthere is a difference due to the different masses and two-body binding momenta of the scatteredparticles which affects the momentum independent prefactors as well as the integrand. However,the numerical treatment of the (coupled) integral equations is universal in the sense that the E,k, p and q dependence of the amplitudes is always of the same form. The numerical implemen-tation used to solve these equations follows the method described in Ref. [79].Firstly, one observes that each integral equation holds for all E and thus k. Therefore we willsuppress their dependence and treat them as parameters which leads to integral equations of theform
T (p) = R(p) +
∫ ∞
0
dq K(p, q)T (q) , (G.1)
where the kernel K(p, q) could contain a pole in q depending on the center-of-mass energy and
the binding momentum γ. Thus, we write it as K(p, q) = K(p, q)/(q − q0 − iε). The positionof the pole (i.e. q0) can be found by rewriting the scattering amplitudes. It is different in eachscattering process, but does not depend on the specific scattering channel. Furthermore, one hasto replace the upper integration boundary by a momentum cutoff Λ in numerical calculations.To get rid of the iε prescription in the denominator of K(p, q) we carry out the limit ε→ 0 usingthe Sokhatsky-Weierstrass theorem,
1
q − q0 ± iε= PV
1
q − q0
∓ iπ δ(q − q0) , (G.2)
205
with the principal value PV. This changes the integral equation Eq. (G.1) to
T (p) = R(p) +
∫ Λ
0
dqK(p, q) T (q)−K(p, q0) T (q0)
q − q0
+K(p, q0) T (q0)
(iπ + ln
[Λ− q0
q0
]),
(G.3)
where the principal value integral is already evaluated:
PV
∫ Λ
0
dq1
q − q0
= ln
[Λ− q0
q0
]. (G.4)
Eq. (G.3) can be discretized by replacing the integration over q by a summation over discretemomenta q =: pj with weights wj and setting pi := p. Introducing the notation Ri := R(pi),Ti := T (pi) and Kij := K(pi, pj) one could write Eq. (G.3) in a discretized form, but as thereare poles in the integral kernel one has to take into account their contribution, too. For this onehas to distinguish three cases, corresponding to the number of poles appearing in the (coupled)integral equations.In the trivial case without pole the procedure is straightforward. One uses N mesh pointsgenerated e.g. by the Gauss-Legendre quadrature [189] to numerically calculate the integral
Ti = Ri +N∑
j=1
wj Kij Tj for i = 1, ..., N , (G.5)
which holds for all i = 1, ..., N and thus can be described by a matrix relation
T = R +M(0)1 T , (G.6)
with(M(0)
1
)ij
= wjKij being a N ×N matrix.
If there is one pole we let N−1 be the number of mesh points and set pN := q0 with corresponding”weight“ wN = 1 which allows us to write Eq. (G.3) as
Ti = Ri +N−1∑
j=1
wjKij Tj −KiN TN
pj − pN+KiN TN
(iπ + ln
[Λ− pNpN
])
= Ri +N−1∑
j=1
wj Kij Tj +KiN TN
(iπ + ln
[Λ− pNpN
]−
N−1∑
k=1
wkpk − pN
), for i = 1, ..., N .
(G.7)
Note, that for i = N there is no special treatment necessary. One has just added an extra meshpoint with a fixed value. Defining a N ×N matrix M(1)
1 ,
(M(1)
1
)ij
=
wjKij , for j < N
wj Kij
(iπ + ln
[Λ−pNpN
]−∑N−1
k=1wk
pk−pN
), for j = N
, (G.8)
206
Eq. (G.7) can be written as simple matrix relation similarly to the case without pole
T = R +M(1)1 T . (G.9)
The third case with two poles only occurs in the coupled integral equations. Thus, one has togeneralize the scheme above to a system of two coupled equations. Consider the following systemof such equations:
T1(p) = R1(p) +
∫ Λ
0
dq K11(p, q) T1(q) +
∫ Λ
0
dq K12(p, q) T2(q) , (G.10)
T2(p) = R2(p) +
∫ Λ
0
dq K21(p, q) T1(q) +
∫ Λ
0
dq K22(p, q) T2(q) , (G.11)
where it is assumed that the kernels Ki1, i = 1, 2, have the same pole at q = q0,1 and also that
Ki2, i = 1, 2, have a common pole at q = q0,2 6= q0,1. Comparing this assumption with the coupledintegral equations in section 5.1 one notices that it is indeed justified. Each integral is discretizedusing the same set of mesh points pj := q with weights wj for j = 1, ..., (N − number of poles).Hence, with the notation introduced above (T1(p) = (T1)i, K12(p, q) = (K12)ij, etc.) one finds
(T1)i = (R1)i +
N−poles∑
j=1
wj (K11)ij (T1)j +
N−poles∑
j=1
wj (K12)ij (T2)j , for i = 1, ..., N (G.12)
(T2)i = (R2)i +
N−poles∑
j=1
wj (K21)ij (T1)j +
N−poles∑
j=1
wj (K22)ij (T2)j , for i = 1, ..., N . (G.13)
These two coupled equations can be combined into one matrix equation of the form T = R +M2 T, where T is a vector with 2N entries,
T =
(T1)1...
(T1)N(T2)1
...(T2)N
, (G.14)
and M2 is a 2N × 2N matrix whose elements (M2)ij are defined by the integral kernels anddepend on the number of poles in those. If there is no pole present in the kernels one findsM2 ≡M(0)
2 which is given in Eq. (G.18).
In case of a pole in Ki1 and no one in Ki2 we define pN := q0,1 and wN := 1 in order to take
care of the pole contribution to the integral. The matrix M2 then has the form of M(1a)2 given
in Eq. (G.19) where
z(pN) =
iπ + ln
[Λ− pNpN
]−
N−poles∑
k=1
wkpk − pN
, (G.15)
207
is the pole contribution. Consequently, the (2N)-th column of M(1a)2 is filled with zeros since
there is no such contribution for the kernels Ki2 which have no pole.If the pole distribution is interchanged (i.e. one pole in Ki2, no pole in Ki1), the extra meshpoint is given by pN := q0,2 with wN = 1 and the structure of the N -th and the (2N)-th column
is interchanged, too. Therefore M2 ≡M(1b)2 which is defined in Eq. (G.20). Note, that z(pN) is
different from that in M(1a)2 because the position of the pole pN is different.
In the last case where both kernels Ki1 and Ki2 have a pole, one has to combine the results forjust one pole in the right way. Since there are now N − 2 mesh points one fills up the N elementset with the additional points pN−1 := q0,1 and pN := q0,2. The ”wights” are both equal to one,i.e. wN−1 = wN = 1. The pole contributions depend as before on the function z(pN) in whichthe sum over k now runs from 1 to N−2 instead of N−1. This leads to the matrix in Eq. (G.21)
which we will call M(2)2 .
Note, that it is in the same way possible to extend the numerical implementation to an arbitrarynumber of coupled integral equations. Finally, we will make some remarks on the distribution ofmesh points. Since the wave function falls off very quickly for large momenta it is useful to havemore mesh points in the region of small momenta and less in that of large momenta instead ofan equidistantly distribution. This leads to a faster convergence in the number of mesh pointsand can be accomplished in the following way: one calculates a set of (N −poles) mesh points xjwith weights ξj in the interval [0, ln(Λ+1)] using the Gauss-Legendre quadrature implementationdescribed in Ref. [189] (note, that one could also use a different quadrature scheme). These meshpoints and weights are used to define the mesh points pj and weights wj in the discussion above:
wj = exp(xj) ξj , (G.16)
pj = exp(xj)− 1 . (G.17)
In the continuum this definition corresponds to the substitution q(x) = ex − 1 in the integral∫ Λ
0dq =
∫ ln(Λ+1)
0exdx with x ∈ [0, ln(Λ + 1)].
208
M(0)2 =
w1(K11)11 · · · wN (K11)1N w1(K12)11 · · · wN (K12)1N...
......
...
w1(K11)N1 · · · wN (K11)NN w1(K12)N1 · · · wN (K12)NNw1(K21)11 · · · wN (K21)1N w1(K22)11 · · · wN (K22)1N
......
......
w1(K21)N1 · · · wN (K21)NN w1(K22)N1 · · · wN (K22)NN
, (G.18)
M(1a)2 =
w1(K11)11 · · · wN−1(K11)1,N−1 wN (K11)1N z(pN ) w1(K12)11 · · · wN−1(K12)1,N−1 0...
......
......
w1(K11)N1 · · · wN−1(K11)N,N−1 wN (K11)NN z(pN ) w1(K12)N1 · · · wN−1(K12)N,N−1 0
w1(K21)11 · · · wN−1(K21)1,N−1 wN (K21)1N z(pN ) w1(K22)11 · · · wN−1(K22)1,N−1 0...
......
......
w1(K21)N1 · · · wN−1(K21)N,N−1 wN (K21)1N z(pN ) w1(K22)N1 · · · wN−1(K22)N,N−1 0
,
(G.19)
M(1b)2 =
w1(K11)11 · · · wN−1(K11)1,N−1 0 w1(K12)11 · · · wN−1(K12)1,N−1 wN (K12)1N z(pN )...
......
......
w1(K11)N1 · · · wN−1(K11)N,N−1 0 w1(K12)N1 · · · wN−1(K12)N,N−1 wN (K12)1N z(pN )
w1(K21)11 · · · wN−1(K21)1,N−1 0 w1(K22)11 · · · wN−1(K22)1,N−1 wN (K22)1N z(pN )...
......
......
w1(K21)N1 · · · wN−1(K21)N,N−1 0 w1(K22)N1 · · · wN−1(K22)N,N−1 wN (K22)1N z(pN )
,
(G.20)
M(2)2 =
w1(K11)11 · · · wN−2(K11)1,N−2 wN−1(K11)1,N−1 z(pN−1) 0...
......
...
w1(K11)N1 · · · wN−2(K11)N,N−2 wN−1(K11)N,N−1 z(pN−1) 0
w1(K21)11 · · · wN−2(K21)1,N−2 wN−1(K21)1,N−1 z(pN−1) 0...
......
...
w1(K21)N1 · · · wN−2(K21)N,N−2 wN−1(K21)1,N−1 z(pN−1) 0
w1(K12)11 · · · wN−2(K12)1,N−2 0 wN (K12)1,N z(pN )...
......
...
w1(K12)N1 · · · wN−2(K12)N,N−2 0 wN (K12)1,N z(pN )
w1(K22)11 · · · wN−2(K22)1,N−2 0 wN (K22)1,N z(pN )...
......
...
w1(K22)N1 · · · wN−2(K22)N,N−2 0 wN (K22)1,N z(pN )
. (G.21)
209
Bibliography
[1] M. Peskin, D. Schroeder, ”An Introduction to Quantum Field Theory“, Westview Press,1995.
[2] K. A. Olive et al. (Particle Data Group), Chin. Phys. C, 38, 090001 (2014).
[3] D. Griffiths, ”Introduction to Elementary Particle Phyiscs“, 2nd ed., Wiley-VCH VerlagGmbH & Co. 2009.
[4] R. L. Jaffe, J. Johnson, Phys. Lett. B 60 (1976) 201-204.
[5] D. Robson, Nucl. Phys. B 130 (1977) 328-348.
[6] V. Mathieu, N. Kochelev, V. Vento, Int. J. Mod. Phys. E 18 (2009) 1-49.
[7] M. Chanowitz, S. Sharpe, Phys. Lett. B 222 (1983) 211-244.
[8] R. L. Jaffe, Phys. Rev. D 15, 281 (1977).
[9] L. Maiani, V. Riquer, F. Piccinini, A. D. Polosa, Phys. Rev. D 71, 014028 (2005).
[10] L. Maiani, V. Riquer, F. Piccinini, A. D. Polosa, Phys. Rev. D 72, 031502 (2005).
[11] L. Maiani, A.D. Polosa, V. Riquer, arXiv:0708.3997.
[12] M. B. Voloshin, L. B. Okun, Pis’ma Zh. Eksp. Teor. Fiz. 23 369 (1976), JETP Lett. 23,333 (1976).
[13] A. De Rujula, H. Georgi, S. L. Glashow, Phys. Rev. Lett. 38 317 (1977).
[14] N. A. Tornqvist, Phys. Rev. Lett. 67 556 (1991).
[15] A. Abashian et al. (Belle Collaboration), Nuclear Instruments and Methods in PhysicsResearch A 479 (2002) 117232.
[16] B. Aubert et al. (BaBar Collaboration), Nuclear Instruments and Methods in PhysicsResearch A 479 (2002) 1116.
[17] J.Z. Bai et al. (BES Collaboration), Nuclear Instruments and Methods in Physics ResearchA 344 (1994) 319-334. & D.M. Asner et al. (BES III Collaboration), Int. J. Mod. Phys.A 24 (2009) S1-794.
210
[18] A. Augusto Alves, Jr. et al. (LHCb Collaboration), JINST 3 (2008) S08005.
[19] M. Cleven, F.-K. Guo, C. Hanhart, Q. Wang, Q. Zhao, Phys. Rev. D 92, 014005 (2015).
[20] S.-K. Choi et al. (Belle Collaboration), Phys. Rev. Lett. 91, 262001 (2003).
[21] B. Aubert et al. (BaBar Collaboration), Phys. Rev. Lett. 95, 142001 (2005).
[22] M. Ablikim et al. (BES III Collaboration), Phys. Rev. Lett. 110, 252001 (2013).
[23] Z. Q. Liu et al. (Belle Collaboration), Phys. Rev. Lett. 110, 252002 (2013).
[24] T. Xiao, S. Dobbs, A. Tomaradze, K. K. Seth, Phys. Lett. B 727, 366 (2013).
[25] M. Ablikim et al. (BES III Collaboration), Phys. Rev. Lett. 112, 022001 (2014).
[26] M. Ablikim et al. (BES III Collaboration), Phys. Rev. Lett. 111, 242001 (2013).
[27] M. Ablikim et al. (BES III Collaboration), Phys. Rev. Lett. 112, 132001 (2014).
[28] M. Ablikim et al. (BES III Collaboration), Phys. Rev. Lett. 113, 212002 (2014).
[29] S.-K. Choi et al. (Belle Collaboration), Phys. Rev. Lett. 100, 142001 (2008).
[30] K. Chilikin et al. (Belle Collaboration), Phys. Rev. D 88, 074026 (2013).
[31] R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 112, 222002 (2014).
[32] R. Mizuk et al. (Belle Collaboration), Phys. Rev. D 78, 072004 (2008).
[33] A. Bondar et al. (Belle Collaboration), Phys. Rev. Lett. 108, 122001 (2012).
[34] M. Nielsen, F. S. Navarra, S. H. Lee, Phys. Rep. 497 (2010) 40-83.
[35] E. Braaten, Ch. Langmack, D. Hudson Smith, Phys. Rev. D 90, 014044 (2014).
[36] B. Aubert et al. (BaBar Collaboration), Phys. Rev. Lett. 90, 242001 (2003).
[37] D. Besson et al. (CLEO Collaboration), Phys. Rev. D 68, 032002 (2003).
[38] G. Alexander et al., Phys. Rev. 173 (1968) 1452.
[39] B. Sechi-Zorn et al., Phys. Rev. 175 (1968) 1735-1740.
[40] J. Weinstein, N. Isgur, Phys. Rev. Lett. 48 (1982) 659.
[41] J. Weinstein, N. Isgur, Phys. Rev. D 27 (1983) 588.
[42] J. Weinstein, N. Isgur, Phys. Rev. D 41 (1990) 2236.
[43] T. Barnes, F. Close, H. Lipkin, Phys. Rev. D 68, 054006 (2003).
211
[44] E. van Beveren, G. Rupp, Phys. Rev. Lett. 91, 012003 (2003).
[45] F.-K. Guo, U.-G. Meißner, Phys. Rev. D 84 (2011) 014013.
[46] L. Liu, K. Orginos, F.K. Guo, Ch. Hanhart, U.-G. Meißner, PoS LATTICE2013 (2014)239.
[47] M. Cleven, H. W. Grießhammer, F.-K. Guo, Ch. Hanhart, U.-G. Meißner, Eur. Phys. J.A 50 (2014) 149.
[48] Z.-H. Guo, U.-G. Meißner, D.-L. Yao, Phys. Rev. D 92, 094008 (2015).
[49] Q. Wang, Ch. Hanhart, Q. Zhao, Phys. Rev. Lett. 111, 132003 (2013).
[50] F.-K. Guo, C. Hidalgo-Duque, J. Nieves, M. Pavon Valderrama, Phys. Rev. D 88, 054007(2013).
[51] M. Nielsen, F. S. Navarra, S. H. Lee, Phys. Rep. 497 (2010) 40-83.
[52] X. Liu, Z.-G. Luo, Y. R. Liu, S.-L. Zhu, Eur. Phys. J. C 61, 411-428 (2009).
[53] G.-J. Ding, Phys. Rev. D 79, 014001 (2009).
[54] X. Liu, Y.-R. Liu, W.-Z. Deng, S.-L. Zhu, Phys. Rev. D 77, 034003 (2008).
[55] A. E. Bondar, A. Garmash, A. I. Milstein, R. Mizuk, M. B. Voloshin, Phys. Rev. D 84,054010 (2011).
[56] R. H. Dalitz, S. F. Tuan, Phys. Rev. Lett. 2 (1959) 425-428.
[57] R. H. Dalitz, S. F. Tuan, Annals Phys. 10 (1960) 307-351.
[58] M. B. Voloshin, Phys. Lett. B 579 (2004) 316-320.
[59] J. Brodzicka et al. (Belle Collaboration), Phys. Rev. Lett. 100, 092001 (2008).
[60] J. A. Oller, E. Oset, Nucl. Phys. A 620 (1997) 438-456 & Nucl. Phys. A 652 (1999) 407.
[61] G. Janssen, B. C. Pearce, K. Holinde, J. Speth, Phys. Rev. D 52 (1995) 2690-2700.
[62] V. Baru, J. Haidenbauer, C. Hanhart, Yu. Kalashnikova, A. E. Kudryavtsev, Phys. Lett.B 586 (2004) 53-61.
[63] E. Klempt, A. Zaitsev, Phys. Rep. 454 (2007) 1-202.
[64] S. Weinberg, Physica 96 A (1979) 327-340.
[65] S. Weinberg, Phys. Lett. B 251 (1990) 288-292.
[66] J. Gasser, H. Leutwyler, Annals Phys. 158 (1984) 142-210.
212
[67] J. Gasser, H. Leutwyler, Nucl. Phys. B 250 (1985) 465-516.
[68] V. Bernard, N. Kaiser, U.-G. Meißner, Int. J. Mod. Phys. E 4 (1995) 193-346.
[69] U. van Kolck, Nucl. Phys. A 645 (1999) 273-302.
[70] D. B. Kaplan, M. J. Savage, M. B. Wise, Nucl. Phys. B 478, 629-659 (1996).
[71] D. B. Kaplan, M. J. Savage, M. B. Wise, Phys. Lett. B 424, 390-396 (1998).
[72] D. B. Kaplan, M. J. Savage, M. B. Wise, Nucl. Phys. B 534, 329-355 (1998).
[73] D. B. Kaplan, M. J. Savage, M. B. Wise, Phys. Rev. C 59, 617-629 (1999).
[74] U. van Kolck, Prog. Part. Nucl. Phys. 43, 337 (1999).
[75] S. R. Beane, P. F. Bedaque, W. C. Haxton, D. R. Phillips, M. J. Savage, arXiv:nucl-th/0008064.
[76] P. F. Bedaque, U. van Kolck, Annu. Rev. Nucl. Part. Sci. 52, 339 (2002).
[77] E. Epelbaum, H.-W. Hammer, U.-G. Meißner, Rev. Mod. Phys. 81, 1773-1825 (2009).
[78] D. B. Kaplan, M. J. Savage, R. P. Springer, M. B. Wise, Phys. Lett. B 449 (1999) 1-5.
[79] S. Konig, Diploma thesis, unpublished (2010) & S. Konig, H.-W. Hammer, Few BodySyst. 54 (2013) 231-234.
[80] E. Braaten, H.-W. Hammer, Phys. Rep. 428, 259-390 (2006).
[81] R. Shankar, ”Principles of Quantum Mechanics“, Springer, 1994.
[82] P. G. Burke, ”R-Matrix Theory of Atomic Collisions“, Springer, 2011.
[83] H. A. Bethe, Phys. Rev. 76, 38 (1949).
[84] H. A. Bethe, C. Longmire, Phys. Rev. 77, 647 (1950).
[85] V. Efimov, Phys. Lett. B 33 (1970) 563-564.
[86] R. D. Amado, J. V. Noble, Phys. Lett. B 35 (1971) 25-27.
[87] R. D. Amado, J. V. Noble, Phys. Rev. D 5 (1972) 1992-2002.
[88] F. Ferlaino, R. Grimm, Physics 3, 9 (2010).
[89] D. E. Gonzales Trotter et al., Phys. Rev. Lett. 83 (1999), 37883791.
[90] G. A. Miller, B. M. K. Nefkens, I. Slaus, Phys. Rept. 194, 1116 (1990).
[91] P. F. Bedaque, H.-W. Hammer, U. van Kolck, Phys. Rev. Lett. 82 (1999), 463-467.
213
[92] P. F. Bedaque, H.-W. Hammer, U. van Kolck, Nucl. Phys. A 646 444-466 (1999).
[93] P. F. Bedaque, H.-W. Hammer, U. van Kolck, Nucl. Phys. A 676 357-370 (2000).
[94] G. V. Skorniakov, K. A. Ter-Martirosian, Sov. Phys. JETP 4 (1957) 648.
[95] V. V. Komarov, A. M. Popova, Nucl. Phys. 54 (1964) 278-294.
[96] K.Madison, Y. Wang, A. M. Rey, K. Bongs (Editors), ”Annual Review of Cold Atomsand Molecules: Volume 1“, World Scientific 2013.
[97] H. Feshbach, Ann. Phys. 19 (1962), 287-313.
[98] E. Tiesinga, A. J. Moerdijk, B. J. Verhaar, H. T. C. Stoof, Phys. Rev. A 46 (1992),R1167-R1170.
[99] E. Tiesinga, B. J. Verhaar, H. T. C. Stoof, Phys. Rev. A 47 (1993), 4114-4122.
[100] T. Kraemer, M. Mark, P. Waldburger, J.G. Danzl, C. Chin, B. Engeser, A. D. Lange, K.Pilch, A. Jaakkola, H.-C. Nagerl, R. Grimm, Nature 440, 315 (2006).
[101] S. Knoop, F. Ferlaino, M. Mark, M. Berninger, H. Schobel, H.-C. Nagerl, R. Grimm,Nature Phys. 5, 227 (2009).
[102] M. Zaccanti, B. Deissler, C. D’Errico, M. Fattori, M. Jona-Lasinio, S. Muller, G. Roati,M. Inguscio, G. Modugno, Nature Phys. 5, 586 (2009).
[103] N. Gross, Z. Shotan, S. Kokkelmans, L. Khaykovich,Phys. Rev. Lett. 103, 163202 (2009).
[104] S. E. Pollack, D. Dries, R. G. Hulet, Science 18, 1683 (2009).
[105] T. B. Ottenstein, T. Lompe, M. Kohnen, A. N. Wenz, S. Jochim, Phys. Rev. Lett. 101,203202 (2008).
[106] J. H. Huckans, J. R. Williams, E. L. Hazlett, R. W. Stites, K. M. O’Hara, Phys. Rev.Lett. 102, 165302 (2009).
[107] T. Lompe, T. B. Ottenstein, F. Serwane, A. N. Wenz, G. Zurn, S. Jochim, Science 330,940 (2010).
[108] S. Nakajima, M. Horikoshi, T. Mukaiyama, P. Naidon, M. Ueda, Phys. Rev. Lett. 106,143201 (2011).
[109] B. D. Esry, C. H. Greene, J. P. Burke, Phys. Rev. Lett. 83, 1751 (1999).
[110] E. Braaten, H.-W. Hammer, Phys. Rev. Lett. 87, 160407 (2001).
[111] E. Braaten, H.-W. Hammer, Phys. Rev. A 70, 042706 (2004).
[112] K. Riisager, Rev. Mod. Phys. 66, 1105 (1994).
214
[113] M. V. Zhukov, B. V. Danilin, D. V. Fedorov, J. M. Bang, I. J. Thompson, J. S. Vaagen,Phys. Rep. 231, 151 (1993).
[114] P. G. Hansen, A. S. Jensen, B. Jonson, Ann. Rev. Nucl. Part. Sci. 45, 591 (1995).
[115] A. S. Jensen, K. Riisager, D. V. Fedorov, E. Garrido, Rev. Mod. Phys.76, 215 (2004).
[116] C. A. Bertulani, H.-W. Hammer, U. Van Kolck, Nucl. Phys. A 712, 37 (2002).
[117] P.F. Bedaque, H.-W. Hammer, U. van Kolck, Phys. Lett. B 569, 159 (2003).
[118] L. Platter, Few-Body Syst. 46, 139 (2009).
[119] H.-W. Hammer, L. Platter, Ann. Rev. Nucl. Part. Sci. 60, 207 (2010).
[120] K. Riisager, ”Nobel Symposium 152: Physics with Radioactive Beams“, Phys. Scr. T 152(2013) 014001.
[121] D. V. Fedorov, A. S. Jensen, K. Riisager, Phys. Rev. Lett. 73, 2817 (1994).
[122] A. E. A. Amorim, T. Frederico, L. Tomio, Phys. Rev. C 56, R2378 (1997).
[123] I. Mazumdar, V. Arora, V. S. Bhasin, Phys. Rev. C 61, R051303 (2000).
[124] M. T. Yamashita, L. Tomio, T. Frederico, Nucl. Phys. A 735, 40 (2004).
[125] D. L. Canham, H.-W. Hammer, Eur. Phys. J. A 37, 367 (2008).
[126] D. L. Canham, H.-W. Hammer, Nucl. Phys. A 836, 275 (2010).
[127] B. Acharya, C. Ji, D. R. Phillips, Phys. Lett. B 723, 196 (2013).
[128] T. Frederico, A. Delfino, L. Tomio, M. T. Yamashita, Prog. Part. Nucl. Phys. 67, 939(2012).
[129] E. Braaten, P. Hagen, H.-W. Hammer, L. Platter, Phys. Rev. A 86, 012711 (2012).
[130] B. Sechi-Zorn et al., Phys. Rev. 175 (1968) 1735.
[131] G. Alexander et al., Phys. Rev. 173 (1968) 1452.
[132] D. L. Canham, H.-W. Hammer, R. P. Springer, Phys. Rev. D 80, 014009 (2009).
[133] E. Braaten, M. Kusunoki, Phys. Rev. D 69, 074005 (2004).
[134] M. Cleven, Q. Wang, F.-K. Guo, Ch. Hanhart, U.-G. Meißner, Q. Zhao, Phys. Rev. D 87,074006 (2013)
[135] Q. Wang, M. Cleven, F.-K. Guo, Ch. Hanhart, U.-G. Meißner, X.-G. Wu, Q. Zhao, Phys.Rev. D 89 (2014) 034001.
215
[136] M. Cleven, Q. Wang, F.-K. Guo, Ch. Hanhart, U.-G. Meißner, Q. Zhao, Phys. Rev. D 90(2014) 074039.
[137] F.-K. Guo, Ch. Hanhart, U.-G. Meißner, Phys. Rev. Lett. 102 (2009) 242004.
[138] F.-K. Guo, L. Liu, U.-G. Meißner, P. Wang, Phys. Rev. D 88 (2013) 074506.
[139] M. Cleven, F.-K. Guo, Ch. Hanhart, U.-G. Meißner, Eur. Phys. J. A 47 (2011) 120.
[140] F.-K. Guo, Ch. Hanhart, Q. Wang, Q. Zhao, Phys. Rev. D 91 (2015) 051504.
[141] Q. Wang, Ch. Hanhart, Q. Zhao, Phys. Rev. Lett. 111 (2013) 132003.
[142] E. Braaten, arXiv:hep-ph/1310.1636 (2013).
[143] E. Braaten, M. Lu, Phys. Rev. D 79 (2009) 051503.
[144] E. Braaten, M. Lu, Phys. Rev. D 77 (2008) 014029.
[145] E. Braaten, M. Lu, Phys. Rev. D 76 (2007) 094028.
[146] E. Braaten, M. Lu, J. Lee, Phys. Rev. D 76 (2007) 054010.
[147] E. Braaten, Ch. Langmack, D. Hudson Smith, Phys. Rev. D 90 (2014) 014044.
[148] E. Braaten, Ch. Langmack, D. Hudson Smith, Phys. Rev. Lett. 112 (2014) 222001.
[149] M. Jansen, H.-W. Hammer, Y. Jia, arXiv:hep-ph/1505.04099 (2015).
[150] E. Wilbring, H.-W. Hammer, U.-G. Meißner, Phys. Lett. B 726 (2013) 326-329.
[151] T. Mehen, Int. J. Mod. Phys. Conf. Ser. 35 (2014) 1460431.
[152] T. Mehen, J. Powell, Phys. Rev. D 84 (2011) 114013.
[153] T. Mehen, J. Powell, Phys. Rev. D 88 (2013) 034017.
[154] A. Margaryan, R. P. Springer, Phys. Rev. D 88 (2013) 014017.
[155] M. B. Voloshin, arXiv:hep-ph/0605063 (2006).
[156] V. Kubarovsky, M. B. Voloshin, Phys. Rev. D 92 (2015) 031502.
[157] X. Li, M. B. Voloshin, Phys. Rev. D 91 (2015) 114014.
[158] X. Li, M. B. Voloshin, Phys. Rev. D 90 (2014) 014036.
[159] M. B. Voloshin, Phys. Rev. D 87 (2013) 091501.
[160] M. B. Voloshin, Phys. Rev. D 87 (2013) 074011.
[161] X. Li, M. B. Voloshin, Phys. Rev. D 86 (2012) 077502.
216
[162] M. B. Voloshin, Phys. Rev. D 86 (2012) 034013.
[163] J. D. Jackson, ”Classical Electrodynamics“, 3rd ed., John Wiley & Sons, Inc., 1999.
[164] I. S. Gradshteyn, I. M. Ryzhik, ”Table of Integrals, Series and Products“, 5th ed., Aca-demic Press, San Diego, 1994.
[165] H.W. Grießhammer, Nucl. Phys. A 760 (2005) 110138.
[166] G. S. Danilov, Sov. Phys. JETP 13 (1961) 349.
[167] H.-W. Hammer, Nucl. Phys. A 705 173-189 (2002).
[168] P. M. Morse, H. Feshbach, ”Methods of Theoretical Physics, Part I“, McGrawHill, NewYork, 1953.
[169] E. P. Wigner, ”On matrices which reduce Kronecker products of representations of S. R.groups“, unpublished, (1951).
[170] A. R. Edmonds, ”Angular Momentum in Quantum Mechanics“, 1st edition, PrincetonUniversity Press, Princeton, 1957.
[171] J. Kolorenc, ”Wick contractions in LATEX with simplewick.sty“ (2006), availableat: http://bay.uchicago.edu/tex-archive/macros/latex/contrib/simplewick/
(access date: October 2015).
[172] V. Efimov, Nucl. Phys. A 210, 157 (1973).
[173] K. Helfrich, H.-W. Hammer, J. Phys. B44 215301 (2011).
[174] U.-G. Meißner, private communication.
[175] R. L. Jaffe, Phys. Rev. Lett. 38 (1977) 195; R. L. Jaffe, Phys. Rev. Lett. 38 (1977) 617,Erratum.
[176] D. Jido, J. A. Oller, E. Oset, A. Ramos, U.-G. Meißner, Nucl. Phys. A 725 (2003) 181-200.
[177] D. Jido, J. A. Oller, E. Oset, A. Ramos, U.-G. Meißner, Nucl.Phys. A755 (2005) 669-672.
[178] Y. Nogami, Phys. Lett. 7, 288 (1963).
[179] T. Yamazaki, Y. Akaishi, Phys. Lett. B 535, 70 (2002).
[180] N. V. Shevchenko, J. Revai, Phys. Rev. C 90, 034003 (2014).
[181] R. Munzer et al., Hyperfine Interactions August 2015, Volume 233, Issue 1, pp 159-166.
[182] T. Hashimoto et al. (J-PARC E15 Collaboration), Prog. Theor. Exp. Phys. (2015) 061D01.
[183] J. Haidenbauer, U.-G. Meißner, Phys. Lett. B 706 (2011) 100105.
217
[184] J. Haidenbauer, U.-G. Meißner, S. Petschauer, Eur. Phys. J. A 51 (2015) 17.
[185] F. Etminan et al. (HAL QCD Collaboration), Nucl. Phys. A 928 (2014) 89-98.
[186] M. Yamada et al. (HAL QCD Collaboration), Prog. Theor. Exp. Phys. (2015) 071B01.
[187] S. Fleming, M. Kusunoki, T. Mehen, U. van Kolck, Phys. Rev. D 76, 034006 (2007).
[188] E. Braaten, H.-W. Hammer, T. Mehen, Phys. Rev. D 82, 034018 (2010).
[189] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, ”Numerical Recipes inC“, 2nd ed., Cambridge University Press (2002).
[190] E. Anderson et al., ”LAPACK Users’ Guide“, 3rd ed., Society for Industrial and AppliedMathematics (1999).
218
Acknowledgments
First of all I want to thank Professor Dr. Hans-Werner Hammer for the opportunity to writemy master thesis as part of his working group and for the supervision of this work. I alsothank Professor Dr. Ulf-G. Meißner for being the co-referee of my thesis and for the opportunityto finish my PhD in Bonn after the departure of Professor Dr. H.-W. Hammer to the TUDarmstadt. Furthermore, I want to express my gratitude to the third and fourth member of thePhD-committee, Professor Dr. Reinhard Beck and Professor Dr. Clemens Simmer.For the support in all the official things one has to care about as a PhD student and also forsome nice small talk every now and then I want to thank Barbara Kraus and Christa Borsch.For the time they spend to proofread this thesis I thank my friends Dr. Stephanie Drosteand Bastian Wurm. For many useful and helpful discussions about Efimov physics I thank inparticular Dr. Sebastian Konig and Dr. Philipp Hagen. Moreover, I thank my (former andpresent) office mates Dr. Daniela Morales Tolentino Zang, Dr. Dominik Hoja, Dr. MichaelLage, Dr. Jacobo Ruiz de Elvira and Dr. Guillermo Rios Marquez on the one hand for manyhours of focused working and on the other hand for useful discussions about physics and a lotof nice conversations about many other topics besides physics. Last but not least, I thank allother members of the theory working group, especially Johanna Daub, Franz Niecknig, Dr. MarkLenkewitz, Matthias Frink, Nico Klein and Dr. Shahin Bour for a great time at the HISKP.
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